Nearly Optimal Average-Case Complexity of Counting Bicliques Under SETH
NNearly Optimal Average-Case Complexity of Counting BicliquesUnder SETH
Shuichi Hirahara ∗ and Nobutaka Shimizu † October 13, 2020
Abstract
In this paper, we seek a natural problem and a natural distribution of instances such thatany O ( n c − (cid:15) ) time algorithm fails to solve most instances drawn from the distribution, whilethe problem admits an n c + o (1) -time algorithm that correctly solves all instances. Specifically,we consider the K a,b counting problem in a random bipartite graph, where K a,b is a completebipartite graph and a and b are constants. Our distribution consists of the binomial randombipartite graphs B αn,βn with edge density 1 /
2, where α and β are drawn uniformly at randomfrom { , . . . , a } and { , . . . , b } , respectively. We determine the nearly optimal average-casecomplexity of this counting problem by proving the following results. Conditional Tight Worst-Case Complexity.
Under the Strong Exponential Time Hypoth-esis, for any constants a ≥ (cid:15) >
0, there exists a constant b = b ( a, (cid:15) ) such that no O ( n a − (cid:15) )-time algorithm counts the number of K a,b subgraphs in a given n -vertex graph.On the other hand, for any constant a ≥ b = b ( n ), we can count all K a,b subgraphs in time bn a + o (1) . Worst-to-Average Reduction.
If there exists a T ( n )-time randomized heuristic algorithmthat solves the K a,b subgraph counting problem on a random graph B αn,βn with successprobability 1 − / polylog( n ), then there exists a T ( n )polylog( n )-time randomized algorithmthat solves the K a,b subgraph counting problem for any input with success probability 2 / Fine-Grained Hardness Amplification.
Suppose that there is a T ( n )-time algorithm withsuccess probability n − (cid:15) that computes the parity of the number of K a,b subgraphs in H ,where H := G (cid:93) · · · (cid:93) G k is the disjoint union of k = O ( (cid:15) log n ) i.i.d. random graphs G , . . . , G k each of which is drawn from the distribution of B αn,βn . Then there is a T ( n ) n O ( (cid:15) ) -time randomized algorithm that counts K a,b subgraphs for any input with suc-cess probability 2 / colorful subgraphs . For the first result, we reducethe k -Orthogonal Vectors problem to the colorful K a,b detection problem. In the second result,we establish a worst-case-to-average-case reduction for a colorful subgraph counting problembased on the binary-extension technique given by [Boix-Adser`a, Brennan, and Bresler; FOCS19].Then, we reduce colorful K a,b counting to K a,b counting. Regarding the third result, we provethe classical XOR lemma and the direct product theorem in the fine-grained setting for subgraphcounting problems. The core of the proof is an O (log n )-round doubly-efficient interactive proofsystem for the colorful subgraph counting problem such that the honest prover is asked to solvepolylog( n ) instances of the counting problem. The new protocol improves the known interactiveproof system for the t -clique counting problem given by [Goldreich and Rothblum; FOCS18] interms of query complexity. ∗ National Institute of Informatics. Email: s [email protected] † The University of Tokyo. Email: nobutaka [email protected] a r X i v : . [ c s . CC ] O c t ontents EMB ( H )col EMBCOL n,H,q ( · ) over U ( H ) n ( F q ) . . . . . . . . . . 154.2 Step 2: Reduce EMBCOL n,H,q ( U ( H ) n ( F q )) to EMBCOL n,H,q ( U ( H ) n ( { , } )) . . . . . . . 16 EMB ( K a,b ) K a,b -DETECTION . . . . . . . . . . . . . . . . . . 215.3 ETH-Hardness of COLORFUL K a,a -DETECTION . . . . . . . . . . . . . . . . . . . 235.4 An n a + o (1) -Time Algorithm for EMB ( K a,b ) . . . . . . . . . . . . . . . . . . . . . . . 245.5 Reduce ( EMB ( K a,b )col , G ( K a,b ) n, / ) to ( EMB ( K a,b ) , K a,b,n ) . . . . . . . . . . . . . . . . . . 255.6 Proofs of Theorems 2.1 and 2.7 and Proposition 2.3 . . . . . . . . . . . . . . . . . . 25 IP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2.1 Running Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2.2 Completeness and Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2.3 Ability of an Honest Prover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Interactive Proof System for EMB ( K a,b ) . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊕ EMB ( H )col . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.2 Application 2: ⊕ K a - Subgraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Introduction
Understanding the average-case complexity of a computational problem is a fundamental questionin the theory of computation. One main reason is that average-case hardness of a problem can serveas the first step towards building secure cryptographic primitives. Besides, average-case complexityreflects the actual performance of an algorithm in the real world.Motivated by a practical performance analysis, the average-case complexity on a natural distri-bution has gathered special attention. A particular focus has been on the complexity of problems onrandom graphs. It is known that several graph problems such as
Hamiltonian Cycle and
GraphIsomorphism , which are believed not to be in P , can be solved with high probability in polyno-mial time if the input is a random graph [FM97]. Even for problems in P , similar gaps betweenaverage- and worst-case complexity have been observed. For example, finding a maximum match-ing in an unweighted m -edge n -vertex graph admits an O ( m √ n )-time algorithm [MV80], whileit admits an (cid:101) O ( m )-time algorithm that works on random graphs with high probability [Mot94]. Karp [Kar76] proposed the problem of finding a clique in a random graph: He conjectured that onecannot find a clique of size (1 + (cid:15) ) log n in an n -vertex Erd˝os-R´enyi random graph of density 1 / (cid:15) > Proof of Work (PoW). A PoWsystem, introduced by Dwork and Naor [DN92], is a proof system that checks whether an untrustedparty has consumed some amount of computational resources. Such a proof system is useful forcoping with a denial-of-service attack; more recently it has been playing an important role indecentralized cryptocurrencies such as Bitcoin. However, the criticism has been made that thecurrent Bitcoin system consumes considerable amounts of the computational resources of the minersfor what is a meaningless task (i.e., computing the SHA-256 hash). To cope with this issue, Ball,Rosen, Sabin, and Vasudevan [BRSV17b] suggested a framework in which the task of a prover issome meaningful one such as machine learning and finding primes. However, there are few naturalproblems whose average-case complexity is well understood, which makes it difficult to construct aPoW system based on meaningful computational tasks. The aim of this paper is to find a natural computational task whose worst- and average-casecomplexity has a “sharp threshold”. Specifically, for a threshold parameter c , we seek a naturalproblem and a natural distribution of instances such that any n c − o (1) time algorithm fails to solve most instances drawn from the distribution, while the problem admits an n c + o (1) -time algorithmthat correctly solves all instances. A problem which exhibits a sharp threshold between worst-and average-case complexity is required to construct a PoW. The existence of such an artificial problem can be shown by using an average-case version of the time hierarchy theorem (withoutrelying on any unproven assumption) [Wil83, GW00]. However, prior to our work, no natural problem whose sharp threshold between worst-case complexity and average-case complexity on a natural distribution is determined under a widely investigated hypothesis.As a natural computational task, we consider the problem of counting the number of bicliques(also known as complete bipartite graphs) on a natural distribution. To state the problem more The notation (cid:101) O ( · ) hides a polylog( n ) factor. The original paper [BRSV17b] introduced the notion of
Proof of Useful Work . However, it turned out that ana¨ıve protocol satisfies their definition [BRSV18], and it remains an open question to formulate the ideas of proof ofuseful work in a theoretically meaningful way. One approach to cope with this issue is to reduce meaningful computational tasks to an artificial problem whoseaverage-case complexity is well understood, as suggested in [BRSV17b]; however, such a reduction is not necessarilyefficient. Roughly speaking, a PoW system requires an honest prover to solve the problem by using a worst-case solver intime n c + o (1) , while a cheating prover running in time n c − o (1) fails to solve the problem on most instances. H , let Emb ( H ) denote the problem that asks the number of embeddingsof H in G for a given graph G . (An embedding of H in G is an injective homomorphism from H to G ; see Section 3 for the definition of homomorphisms.) The problem Emb ( H ) is equivalent tothe H -subgraph counting problem: The number of H -subgraphs in G is equal to the number ofembeddings of H in G divided by the number of automorphisms of H . Our main interest is thecomplexity of Emb ( K a,b ) , where a and b are constants and K a,b is the complete bipartite graphwith a left vertices and b right vertices.The topic of finding or counting bicliques has been investigated from both practical and theo-retical motivations. On the practical side, this study has applications in data mining [AS94, MT17]and bioinformatics [DAB + a ≥
8, our results determine a sharp threshold between worst-case complexityand average-case complexity of counting the number of bicliques K a,b on a natural distribution fora large constant b , under the Strong Exponential Time Hypothesis (SETH) of Impagliazzo, Paturiand Zane [IPZ01]. We consider the following distribution. Definition 1.1 (Random Bipartite Graph K a,b,n ) . For given parameters a, b, n ∈ N , choose α, β uniformly at random from { , . . . , a } and { , . . . , b } , respectively. Let K a,b,n be the distribution ofa random bipartite graph with nα left vertices and nβ right vertices, where each possible edge isincluded independently with probability / . Our first result determines a (not sharp) threshold between worst- and average-case complexityof (
Emb ( K a,b ) , K a,b,n ) under SETH. Theorem 1.2 (Worst- and Average-Case Complexity of Counting K a,b ) . The following holds. • For any constant a ≥ and any constant b , there exists a worst-case solver running in time n a + o (1) that solves Emb ( K a,b ) . • Under SETH , for any constants (cid:15) > and a ≥ , there exists a constant b = b ( a, (cid:15) ) such thatno average-case solver running in time n a − (cid:15) solves Emb ( K a,b ) on a graph drawn from K a,b,n with success probability greater than − (1 / log n ) C , where C = C ( a, b, (cid:15) ) is a sufficiently largeconstant. Theorem 1.2 is the first result that determines the nearly optimal average-case complexityof natural distributional problems under the widely investigated hypothesis. This result providesinsight towards understanding the hardest instance of subgraph counting problems.However, there still remain two issues. The first issue is that the success probability of theaverage-case solver mentioned in Theorem 1.2 is required to be high. As a consequence, the thresh-old between worst- and average-case complexity is not sharp enough. The second issue is that thesubgraph K a,b is not fixed in Theorem 1.2: The parameter b depends on (cid:15) . The problem of countingfixed K a,b might be more natural in the context of complexity theory.To deal with the first issue, we amplify the average-case hardness by considering the problemof solving multiple instances of the K a,b subgraph counting problem. Let ( K a,b,n ) k denote thedistribution of k random graphs drawn independently from K a,b,n . The following result exhibits a sharp threshold between worst- and average-case complexity of the “ k -wise direct product” of thedistributional problem ( Emb ( K a,b ) , K a,b,n ). Strictly speaking, we assume a variant of SETH that rules out (2 − (cid:15) ) n -time randomized algorithms solving k -SATfor some constant k ≥ k ( (cid:15) ). heorem 1.3 (Average-Case Complexity of Counting K a,b for Multiple Instances) . Under SETH,for any constants (cid:15) > and a ≥ , there exists a constant b = b ( a, (cid:15) ) such that no average-casesolver running in time n a − O ( (cid:15) ) can count the numbers of K a,b subgraphs in all the k graphs drawnfrom the direct product ( K a,b,n ) k with success probability greater than n − (cid:15) , where k = O ( (cid:15) log n ) . Our proof techniques of amplifying average-case hardness can be applied to other subgraphcounting problems. Consider the problem ⊕ K a - Subgraph of asking the parity of the number of K a subgraphs contained in a given input graph, where K a denotes the clique of size a . Let G n, / denotethe distribution of the Erd˝os-R´enyi random graph G ( n, / (cid:85) k G n, / denote the distribu-tion of the disjoint union of k random graphs G , . . . , G k each of which is independently drawn from G n, / . We show that the distribution (cid:85) k G n, / is a “hardest” distribution for ⊕ K a - Subgraph ,by presenting an error-tolerant worst-case-to-average-case reduction from ⊕ K a - Subgraph to thedistributional problem ( ⊕ K a - Subgraph , (cid:85) k G n, / ). Theorem 1.4 (Worst-Case to Average-Case Reduction for ⊕ K a - Subgraph ) . Let (cid:15) > and a ∈ N be arbitrary constants. If there exists a T ( n ) -time randomized heuristic algorithm that solves ⊕ K a - Subgraph on a graph drawn from (cid:85) k G n, / with success probability greater than + n − (cid:15) forany k = O ( (cid:15) log n ) , then there exists a randomized algorithm that solves ⊕ K a - Subgraph on anyinput in time T ( n ) n O ( (cid:15) ) . Since any decision problem can be solved with success probability by outputting a randombit, the success probability of an average-case solver in Theorem 1.4 is nearly optimal. There-fore, Theorem 1.4 shows that the decision problem ⊕ K a - Subgraph exhibits some sharp thresholdbetween worst- and average-case complexity. Boix-Adser`a, Brennan, and Bresler [BABB19] and Goldreich [Gol20] presented worst-case-to-average-case reductions from ⊕ K a - Subgraph to ( ⊕ K a - Subgraph , G n, / ). Their reductions arenot error-tolerant and require the success probability of an average-case solver to be close to 1. Theyleft an open question of improving the error tolerance of the reductions. Theorem 1.4 improves theerror tolerance, albeit for a slightly different distribution.At the heart of the proof of Theorems 1.3 and 1.4 is a doubly-efficient interactive proof systemwith subpolynomial number of queries, which is of independent interest. Theorem 1.5 (Interactive Proof System for K a,b Counting with Subpolynomial Queries) . Thereis an O (log n ) -round interactive proof system for Emb ( K a,b ) such that the verifier runs in time O ( n log n ) and asks the prover to solve Emb ( K a,b ) for polylog n instances, where n is the numberof vertices of the given input graph. To cope with the second issue of Theorem 1.2 (i.e., the complexity of counting fixed K a,b ), weconsider the special case of b = a and obtain the average-case hardness under Exponential TimeHypothesis (ETH). In this paper, we consider a randomized variant of ETH which asserts that any2 o ( n ) -time randomized algorithm fails to solve 3SAT. Theorem 1.6.
Under ETH, any n o ( a ) -time algorithm fails to count the number of K a,a subgraphson more than a (1 − (1 / log n ) C ) fraction of the inputs drawn from K a,a,n , where C = C ( a ) is aconstant that depends on a . ETH k -OVC Colorful K a,b -Detection Emb ( K a,b )col Emb ( K a,b ) (Theorem 2.1) (cid:0) Emb ( K a,b )col , G ( K a,b ) n, / (cid:1) ( Emb ( K a,b ) , K a,b,n )Theorem 2.5Lemma 5.1Theorem 2.4 Proposition 2.6 Figure 1: The organization of the proof of average-cane hardness of
Emb ( K a,b ) . In what follows, we briefly present the ingredients for our results while reviewing the literature.The overall outline of the proof of Theorem 1.2 is illustrated in Figure 1. In Section 2.1, we describethe worst-case complexity of
Emb ( K a,b ) under SETH and ETH. In Section 2.2, we present ouridea for the worst-case-to-average-case reduction for subgraph counting problems. In Section 2.3,we introduce our new doubly-efficient interactive protocols. In Section 2.4, we introduce a generalframework of hardness amplification in fine-grained complexity settings. We review related resultsin Section 2.5. Nearly Optimal Complexity of
EMB ( K a,b ) . Our first step is to determine the nearly optimal worst-case complexity for
Emb ( K a,b ) under SETH by proving the following results. Theorem 2.1.
For any constants (cid:15) > and a ≥ , there exists a constant b such that one cannotsolve Emb ( K a,b ) in time O ( n a − (cid:15) ) unless SETH fails. Theorem 2.2.
If there exists an n o ( a ) -time algorithm that solves Emb ( K a,a ) , then ETH fails. Proposition 2.3. If a ≥ , for any (cid:15) > and b ∈ N , there is an algorithm that solves Emb ( K a,b ) in time O ( bn a + (cid:15) ) . We are not aware of previous results that determine the nearly optimal complexity of subgraphcounting problems, while the fine-grained complexity of many natural problems, including theAll-Pairs Shortest Paths, 3SUM, Orthogonal Vectors, and related problems, has been extensivelyexplored in the research area of hardness in P [Wil15, LPW17]. SETH-Hardness of
EMB ( K a,b ) . Our key idea for showing Theorem 2.1 is to consider
Color-ful K a,b -Detection , which is defined as follows. Let K n be the n -vertex complete graph. Fora graph H , let K n × H denote the tensor product (see Section 3 for the definition). For a graph G ⊆ K n × H , every vertex v = ( u, i ) ∈ V ( G ) is associated with a color c ( v ) := i ∈ V ( H ). We The current fastest algorithm [NP85] of counting K a subgraphs runs in time O ( n ω (cid:100) a/ (cid:101) ) on n -vertex graphs,where ω denotes the matrix multiplication exponent. However, the precise value of ω is not well understood. F ⊆ K n × H is colorful if (cid:83) v ∈ V ( F ) { c ( v ) } = V ( H ). The problem Colorful K a,b -Detection asks, given a pair ( n, G ) of n ∈ N and a graph G ⊆ K n × K a,b , to decide whether G contains a colorful subgraph F that is isomorphic to K a,b .Exploiting the fact that Colorful K a,b -Detection is more “structured” than Emb ( K a,b ) , wefirst present a reduction from k -Orthogonal Vectors ( k - OV ) to Colorful K a,b -Detection for k := a . Since k - OV is known to be SETH-hard for any k ≥ Colorful K a,b -Detection : Theorem 2.4.
For any constants a ≥ and (cid:15) > , there exists a constant b = b ( a, (cid:15) ) ≥ a suchthat Colorful K a,b -Detection cannot be solved in time O ( m a − (cid:15) ) unless SETH fails, where m is the number of edges of the input graph. To complete the proof of Theorem 2.1, we reduce
Colorful K a,b -Detection to Emb ( K a,b ) by using the inclusion-exclusion principle. This technique is well known in the literature of fixed-parameter complexity (see, e.g. [CM14, Cur18]). We will present the detail in Section 5.2. ETH-Hardness of
EMB ( K a,a ) . The problem of finding a complete bipartite graph K a,a in agiven graph has gathered special attention in parameterized complexity. Lin [Lin15, Lin18] provedthat the problem is W[1]-hard when a is a parameter. His proof implies that the problem of finding K a,a does not admit any n o ( √ a ) -time algorithm unless ETH fails. In particular, under ETH, any n o ( √ a ) -time algorithm fails to solve Emb ( K a,a ) .Theorem 2.2 improves this lower bound by ruling out an n o ( a ) -time algorithm under ETH. Akey idea behind this improvement is to take advantage of the structure of counting the number ofembeddings: We first reduce the problem of finding a clique K a of size a (which is known to beETH-hard [CHKX06]) to Colorful K a,a -Detection , and then reduce it to Emb ( K a,a ) by usingthe inclusion-exclusion principle. The latter reduction exploits the structure of counting. Compared to the worst-case hardness, the average-case hardness of subgraph counting problems hasnot been well understood until very recently [GR18a, BABB19]. A recent breakthrough result ofBoix-Adser`a, Brennan, and Bresler [BABB19] shows that the worst-case and average-case complex-ities of counting k -cliques in an n -vertex Erd˝os-R´enyi graph are equivalent up to a polylog( n )-factor.They left as an open question the extension of their results to other subgraph counting problems.In this paper, we investigate their open question under a different setting, which is one of ourkey insights. We consider the problem Emb ( H )col of counting H - colorful subgraphs in a randomgraph drawn from a specific distribution G ( H ) n, / , which we introduce below. For a fixed graph H ,let Emb ( H )col be the problem that asks the number of colorful subgraphs that are isomorphic to H contained in a given graph G ⊆ K n × H . For G ⊆ K n × H , let Emb ( H )col ( G ) be the numberof colorful H -subgraphs contained in G . Equivalently, Emb ( H )col ( G ) is equal to the number ofembeddings of H in G that preserves colors. Here, we say that an embedding φ preserves colors if u = c ( φ ( u )) holds for any u ∈ V ( H ), where c : V ( G ) → V ( H ) is the coloring of G . See Figure 2 foran illustration. For a fixed graph H , let G ( H ) n, / ⊆ K n × H be a random subgraph such that eachedge e ∈ E ( K n × H ) is included independently with probability 1 /
2. The distribution of G ( H ) n, / isdenoted by G ( H ) n, / . We denote by ( Emb ( H )col , G ( H ) n, / ) the distributional problem of solving Emb ( H )col on a random graph drawn from G ( H ) n, / . 5 G G
Figure 2: An example of colorful subgraphs. In this paper, we do not consider the case on theright-hand side (if G ⊆ K n × H , then G contains neither blue-orange nor green-red edges).Generalizing the proof techniques of Boix-Adser`a, Brennan, and Bresler [BABB19], we provethat Emb ( H )col is reducible to the distributional problem ( Emb ( H )col , G ( H ) n, / ). Theorem 2.5 (Worst-Case-to-Average-Case Reduction for
Emb ( H )col ) . Let H be a fixed graph.Suppose that the distributional problem ( Emb ( H )col , G ( H ) n, / ) can be solved by a T ( n ) -time randomizedheuristic algorithm A with success probability − δ , where δ = (log n ) − C and C = C H is a sufficientlylarge constant depending on H .Then, there is a T ( n ) · polylog( n ) -time randomized algorithm B that solves Emb ( H )col for anyinput with success probability / . Moreover, the number of oracle calls of A by B is at most (log n ) O ( | E ( H ) | ) . It should be noted that, Dalirrooyfard, Lincoln, and Vassilevska Williams [DLW20] proved thesame result as Theorem 2.5 (their work is independent to us).Our technical contribution here is to reduce (
Emb ( K a,b )col , G ( K a,b ) n, / ) to ( Emb ( K a,b ) , K a,b,n ) usingthe inclusion-exclusion principle. Proposition 2.6.
Suppose that there is a T ( n ) -time randomized heuristic algorithm that solves ( Emb ( K a,b ) , K a,b,n ) with success probability − δ . Then, there is an O ( ab a + b · T ( n )) -time random-ized heuristic algorithm that solves ( Emb ( K a,b )col , G ( K a,b ) n, / ) with success probability − O ( ab a + b δ ) . Combining Theorem 2.5 and Proposition 2.6, we obtain a worst-case-to-average-case reductionfrom
Emb ( K a,b ) to ( Emb ( K a,b ) , K a,b,n ). Theorem 2.7.
Let ≤ a ≤ b be arbitrary constants. Suppose that there is a T ( n ) -time ran-domized heuristic algorithm that solves ( Emb ( K a,b ) , K a,b,n ) with success probability − δ , where δ = (log n ) − C and C = C ( a, b ) is a sufficiently large constant.Then, there is a T ( n ) · polylog( n ) -time randomized algorithm that solves Emb ( K a,b ) for anyinput with success probability / . Theorem 2.7 is of interest in its own right; we emphasize that a and b can be chosen arbitrarilyunlike Theorem 2.1 (i.e., the SETH-hardness of Emb ( K a,b ) ). In the context of subgraph counting,counting K , (i.e., 4-cycle) subgraphs in a graph on n vertices with m edges attracts particularinterest: The current fastest counting algorithm runs in time O ( n ω ) or O ( m . ) [AYZ97], whereasfinding a K , can be done in time O ( n ) [YZ97] or O ( m . ) [AYZ97]. A central question in thiscontext is whether we can beat the O ( n ω )-time algorithm for the K , -counting problem. Theworst-case-to-average-case reduction given in Theorem 2.7 indicates that a random bipartite graphis essentially the hardest distribution for the K , -counting problem.6 roof Ideas. Our proof of Theorem 2.7 is based on the proof techniques of Boix-Adser`a, Brennan,and Bresler [BABB19]. They first reduced the clique counting problem to the clique countingproblem on partite graphs. Then they encoded the counting problem as a polynomial over a largefinite field and reduced the counting problem to the evaluation of the polynomial at a random point.This is a classical method of local decoding of Reed-Muller codes, which is a standard technique inthe literature of random self-reducibility [Lip91, GS92]. The key part of the proof in [BABB19] isa binary expansion technique , which enables us to reduce the polynomial evaluation to the cliquecounting problem on a random partite graph. Specifically, they reduced the evaluation of F ( x ) fora random point x ∈ F Nq to the evaluation of F ( z ) , . . . , F ( z m ) for m = polylog( q ), where F ( · ) is thepolynomial that encodes the clique counting problem and z i ∈ { , } N is a random binary point.Finally, they reduced the clique counting problem on a random partite graph to the clique countingproblem on an Erd˝os-R´enyi random graph by using an inductive argument.The inductive argument given in [BABB19] does not seem to be easily generalized to subgraphsother than cliques. Instead, we present a simple proof that can be applied to bicliques (as well ascliques) by using the inclusion-exclusion principle, and prove Proposition 2.6.The proof of Theorem 2.5 is given by observing that the binary expansion technique of [BABB19]can be applied to any multivariate polynomial F over F q on N variables x , . . . , x N satisfying thefollowing conditions.1. F is a low-degree polynomial. (The reduction runs in time (log n ) O ( D ) , where D is the totaldegree of F .)2. There is a partition ( E , . . . , E (cid:96) ) of [ N ] such that | E j ∩ I | = 1 holds for every monomial (cid:81) i ∈ I x i and j ∈ [ (cid:96) ].These two conditions motivate us to consider the problem Emb ( H )col of counting colorful subgraphs.The recent work of Dalirrooyfard, Lincoln, and Vassilevska Williams [DLW20] called the polyno-mial with this property strongly (cid:96) -partite and presented a worst-case-to-average-case reduction forevaluating this polynomial, which implies Theorem 2.5 (their work is independent to ours). SeeSection 4.1 for details of the our worst-case-to-average-case reduction. A line of research on interactive proof systems, pioneered by Goldwasser, Micali, and Rack-off [GMR89], revealed the surprising power of interaction. Early studies of interactive proofsystems focused on efficient verification of intractable problems such as
PSPACE -complete prob-lems [LFKN92, Sha92]. A recent line of research (e.g., [GKR15, RRR16, GR18b, GR18a, BRSV18])concerns interactive proof systems for tractable problems, which are called doubly-efficient inter-active proof systems : The goal of a doubly-efficient interactive proof system is to verify a state-ment in almost linear time by interacting with a polynomial-time prover. It is worth mentioningthat a doubly-efficient interactive proof system plays an important role in Proof of Work sys-tems [BRSV17a, BRSV18].Theorem 1.5 provides a doubly-efficient interactive proof system for the K a,b counting problemwith polylog( n ) queries. More generally, for any fixed graph H , we present an interactive proofsystem for the colorful H -subgraph counting problem Emb ( H )col , in which an honest prover isrequired to solve ( Emb ( H )col , G ( H ) n, / ) on average. We often use n to denote the number of vertices of a given graph; thus, “almost linear time (in the input length)”means (cid:101) O ( n ) time in our context. heorem 2.8 (IP for Emb ( H )col ) . Let H be a graph. There is an O (log n ) -round interactive proofsystem IP for the statement “ Emb ( H )col ( G ) = C ” such that, given an input ( G, n, C ) , • The verifier accepts with probability for some prover if the statement is true (perfect com-pleteness), while it rejects for any prover with probability at least / otherwise (soundness). • In each round, the verifier runs in time n (log n ) O ( | E ( H ) | ) and sends (log n ) O ( | E ( H ) | ) instancesof Emb ( H )col to a prover.Furthermore, for any constant L , there exists a constant L = L ( H, L ) such that, if the state-ment is true and the prover has oracle access to a randomized heuristic algorithm that solves ( Emb ( H )col , G ( H ) n, / ) with success probability − (log n ) − L , then the verifier accepts with probability − (log n ) − L . The “Furthermore” part follows the worst-case-to-average-case reduction of Theorem 2.5: Wecan easily modify an honest prover of IP so that the prover is required to solve polylog( n ) instancesof the distributional problem ( Emb ( H )col , G ( H ) n, / ).The salient feature of our interactive proof system is that the amount of communication betweena verifier and a prover is at most polylog( n ) bits; equivalently, the number of queries that a verifiermakes to a prover is at most polylog( n ). This will be important in the next section—where weprove hardness amplification theorems in a fine-grained setting based on an interactive proof systemwhose query complexity is subpolynomially small.The interactive proof system of Theorem 2.8 can be compared with one given by Goldreichand Rothblum [GR18a]. They presented an O (1)-round (cid:101) O ( n )-query doubly-efficient interactiveproof system for Emb ( K k ) . Theorem 2.8 significantly improves the query complexity from (cid:101) O ( n )to polylog( n ), at the cost of increasing the round complexity from O (1) to O (log n ). To explainthe source of our improvement, we review the ideas of [GR18a]: Their interactive proof system isessentially a variant of the sum-check protocol [LFKN92]. They encoded Emb ( K k ) as a polyno-mial over a large finite field and used the following downward self-reducibility of Emb ( K k ) ( G ): Emb ( K k ) ( G ) = (cid:80) i ∈ V ( G ) Emb ( K k − ) ( G − i ), where G − i denotes the graph obtained by removingthe vertex i from G . In each round, the prover sends a polynomial of degree O ( n ) to the verifier.Each coefficient of the polynomial can be computed by calling a Emb ( K k ) solver polylog n times.Overall, the number of queries made by the verifier is O ( n polylog n ). To summarize, the degree ofthe polynomial is the main bottleneck for the query complexity.We improve the query complexity by exploiting a different type of downward self-reducibility.Roughly speaking, at each round, we reduce verifying Emb ( H )col ( G ) for an n -vertex graph G tothe verification of Emb ( H )col ( G ) , . . . , Emb ( H )col ( G m ) for m = polylog( n ), where each G i has n/ Emb ( H )col as a polynomialof degree | E ( H ) | (2 | V ( H ) | −
1) = O (1) for a fixed graph H , thereby reducing the query complexity.The details are presented in Section 6.We mention that the existence of a doubly-efficient interactive proof system with communicationcomplexity polylog( n ) for Emb ( H )col is guaranteed by using the general result of Goldwasser, Kalai,and Rothblum [GKR15]. However, the strategy of an honest prover of their proof system may notbe computed efficiently with Emb ( H )col oracle. We need an interactive proof system in which anhonest prover is simulated with oracle access to Emb ( H )col , as is guaranteed in Theorem 2.8. Thiswill be important for the applications to hardness amplification theorems, as we explain next.8 .4 Fine-Grained Hardness Amplification The error tolerance of the worst-case-to-average-case reduction from
Emb ( K a,b ) to the distribu-tional problem ( Emb ( K a,b ) , K a,b,n ) presented in Theorem 2.7 is not satisfactory: it requires aheuristic algorithm to solve random instances with probability at least 1 − / polylog( n ). Thiscomes from the fact that we use a union bound in order to guarantee that all the polylog( n ) queriesare answered correctly, exactly as in the work of Boix-Adser`a, Brennan, and Bresler [BABB19].They left an open question of increasing the error tolerance of the reduction. We answer the openquestion partially: By using hardness amplification theorems , we increase the error tolerance, atthe cost of modifying the distribution of a random graph (to some other natural distributions).In this section, we present a general framework for amplifying average-case hardness in thefine-grained complexity settings, based on the techniques from “coarse-grained” complexity theory.Specifically, we prove fine-grained complexity versions of hardness amplification theorems for anyproblem f that admits an efficient selector that makes n o (1) queries. Such a selector for f can beconstructed from a doubly-efficient interactive proof system in which (1) the verifier makes at most n o (1) queries and (2) the strategy of an honest prover can be efficiently computed given oracle accessto f . In particular, the interactive proof system of Theorem 2.8 enables us to establish fine-grainedcomplexity versions of hardness amplification theorems for f = Emb ( H )col for any fixed graph H .We explain the details below. A direct product theorem is one of the fundamental hardness amplification results: It states that,if no small circuit can compute f on more than a (1 − δ )-fraction of inputs, then no small circuitcan compute the k -wise direct product f k on a roughly (1 − δ ) k -fraction of inputs. Here, the k -wisedirect product f k of f is defined as f k ( x , . . . , x k ) := ( f ( x ) , . . . , f ( x k )). Our plan is to applya direct product theorem for the function f := Emb ( K a,b ) in order to amplify the average-casehardness of the distributional problem ( Emb ( K a,b ) , K a,b,n ).However, there is an obstacle for applying hardness amplification to uniform computationalmodels (as opposed to non-uniform computational models such as circuits). A standard proof of adirect product theorem can be applied to only non-uniform computational models (cf. the surveyof Goldreich, Nisan, and Wigderson [GNW11]). Impagliazzo, Jaiswal, Kabanets, and Wigder-son [IJKW10] overcame this issue and presented a direct product theorem that is applicable to slightly non-uniform algorithms. Moreover, their direct product theorem is highly optimized andsimplified, and thus it is applicable to the settings of fine-grained complexity.We note that it is impossible to completely get rid of the non-uniformity from a direct producttheorem (if no property of a function f being amplified is used ). In general, a direct product the-orem can be seen as an approximate version of a local-list-decoding algorithm of an error-correctingcode. To be more specific, the function f is encoded as the k -wise direct product f k , and the directproduct theorem can be regarded as a local-list-decoding algorithm of f k : Given a circuit C thatsolves f k on a roughly (cid:15) ≈ (1 − δ ) k fraction of inputs, the local-list-decoding algorithm of [IJKW10]produces a list of candidate circuits C , · · · , C m , one of which is guaranteed to solve f on more thana 1 − δ fraction of inputs, where m = O (1 /(cid:15) ). The non-uniformity refers to the fact that m ≥ (cid:15) ≈ As we will explain in Section 2.4.2, by exploiting a specific property of a function f (i.e., the existence of a selectorfor f ), we can obtain a completely uniform version of a direct product theorem. k -cliques corresponds to computing some low-degree polynomial. Then,they used a low-degree tester and a self-corrector to obtain the correct value from the list of circuits.However, we cannot exploit their technique of using the local list-decoding algorithm since our goalis to obtain a natural average-case hard distribution. Instead, we invoke the direct product theoremof [IJKW10] and then use our doubly-efficient interactive proof system (Theorem 1.5) to identifythe correct circuit. In order to get rid of a small amount of non-uniformity, we make use of a specific property of afunction f . The notion of (oracle) selector, introduced in [Hir15], exactly characterizes the problemfor which a small amount of non-uniformity can be removed (under any relativized world). Forproblems Π (cid:48) and Π, a selector from Π (cid:48) to Π is an efficient algorithm that solves the problem Π (cid:48) given oracle access to two oracles A , A one of which is guaranteed to compute Π. As shown in[Hir15], it is not hard to see that any selector that can identify a correct circuit among two circuitscan be modified to a selector that can identify a correct circuit among many circuits. In lightof this, what is needed for applying the direct product theorem of [IJKW10] is the existence ofa selector from Emb ( K a,b ) to the task of solving the distributional problem ( Emb ( K a,b ) , K a,b,n )with success probability 1 − δ .In the settings of “coarse-grained” complexity [Hir15], it suffices to consider a polynomial-timeselector since polynomial-time algorithms can be composed nicely. However, in the settings offine-grained complexity, one cannot afford even n Ω(1) queries for each candidate circuit, becausesimulating the circuit takes time n a − (cid:15) . The previous interactive proof system given by Goldreichand Rothblum [GR18a] is not efficient enough in terms of the query complexity (cf. Section 2.3)We overcome this difficulty by using the doubly-efficient interactive proof system that makesat most polylog( n ) queries (Theorem 1.5). Roughly speaking, we can construct a selector bysimulating the verifier of an interactive proof system by using the candidate circuit as a prover.More precisely, for a given input x and two circuits C and C , the selector simulates C and C on input x and obtains the two outputs C ( x ) and C ( x ). Then, the selector runs the interactiveproof system to check whether the output is correct. If one of C or C is correct, the verifieraccepts the corresponding output and the selector outputs the accepted one. In this way, by usingTheorem 1.5, we construct a selector as stated in the following theorem. Theorem 2.9 (Selector for
Emb ( H )col Using Subpolynomial Queries) . Let C , . . . , C m be cir-cuits such that, for some i ∗ , the circuit C i ∗ solves ( Emb ( H )col , G ( H ) n, / ) with success probability − (log n ) − K H , where K H is a sufficiently large constant that depends only on H and m =polylog( n ) . Then, there is a randomized n polylog( n ) -time algorithm that, given the list of circuits C , . . . , C m as advice, solves Emb ( H )col correctly with probability at least / by making polylog( n ) queries for each circuit C i . We emphasize the importance of low query complexity of a doubly-efficient interactive proofsystem. Suppose that we can simulate the candidate circuits C and C in time T C ( n ) and that theverifier runs in time T V ( n ), making Q ( n ) queries in the interactive proof system. The running timeof a selector that is constructed from the interactive proof system is roughly O ( T V ( n )+ Q ( n ) T C ( n )).In our setting, T C ( n ) = n a − (cid:15) and thus Q ( n ) must satisfy Q ( n ) = n o (1) .10ombining the “almost uniform” direct product theorem of [IJKW10] with the selector ofTheorem 2.9, we obtain a completely uniform and fine-grained version of a direct product theoremfor the distributional problem ( Emb ( K a,b ) , K a,b,n ), which completes a proof of Theorem 1.3. Let f : { , } n → { , } be a Boolean function. Yao’s XOR lemma asserts that, if no smallcircuit can compute f on more than a (1 − δ ) fraction of inputs, then no small circuit can compute f ⊕ k on a roughly + (1 − δ ) k fraction of inputs, where f ⊕ k : { , } nk → { , } is defined as f ⊕ k ( x , . . . , x k ) := f ( x ) ⊕ · · · ⊕ f ( x k ).An almost uniform version of Yao’s XOR lemma is given by Impagliazzo, Jaiswal, Kabanets,and Wigderson [IJKW10] by combining their direct product theorem with the local list decoding ofthe Hadamard code given by Goldreich and Levin [GL89]. Since the local list decoding algorithmof [GL89] is simple and efficient, we can apply it directly to the fine-grained complexity. As aconsequence, we can prove a uniform and fine-grained version of Yao’s XOR lemma for any problemthat admits an efficient selector.We apply the fine-grained version of Yao’s XOR lemma to the parity variant of Emb ( H )col . Tostate our results formally, let ⊕ Emb ( H )col denote the problem of computing the parity ⊕ Emb ( H )col ( G ) :=( Emb ( H )col ( G ) mod 2) of the number of colorful embeddings of H in a given graph G . Observe that,for k graphs G , . . . , G k ⊆ K n × H , computing ⊕ Emb ( H )col ( G ) ⊕ · · · ⊕ ⊕ Emb ( H )col ( G k ) is equivalent tocomputing ⊕ Emb ( H )col ( G (cid:93) · · · (cid:93) G k ), where F (cid:93) G denotes the disjoint union of two graphs F and G . Let (cid:85) k G ( H ) n, / denote the distribution of G (cid:93) · · · (cid:93) G k , where each G i is independently chosenfrom G ( H ) n, / . Theorem 2.10 (XOR Lemma for ⊕ Emb ( H )col ) . Let H be an arbitrary graph and c > be anarbitrary constant. Suppose that there is a T ( n ) -time randomized heuristic algorithm that solves ( ⊕ Emb ( H )col , (cid:85) k G ( H ) n, / ) for any k = O (log n ) with success probability greater than + n − c . Then,there exists a T ( n ) n O ( c ) -time randomized algorithm that solves ⊕ Emb ( H )col with probability at least / on every input. The proof of Theorem 2.10 is presented in Section 8. The idea is to combine the fine-graineddirect product theorem and the local list decoding of [GL89]. Details can be found in Section 8.
Complexity of Subgraph Counting.
The problem
Emb ( H ) is a fundamental task in thecontext of graph algorithms. For a general subgraph H , we can solve Emb ( H ) in time f ( k ) · n (0 . o (1)) (cid:96) for some function f ( · ), where k and (cid:96) are the number of vertices and edges of H ,respectively [CDM17]. If H has some nice structural property (e.g., small treewidth), severalfaster algorithms are known (see [Cur18] and the references therein). However, to the best ofour knowledge, there is no previous result that precisely determines the complexity of countingsubgraphs. Chen, Huang, Kanj, and Xia [CHKX06] proved that one cannot find a k -clique ina given graph in time f ( k ) · n o ( k ) for any function f ( · ) unless ETH fails. The current fastestalgorithm was given by Nesˇetˇril and Poljak [NP85], who presented an O ( n ω (cid:100) k/ (cid:101) )-time algorithmthat counts the number of k -cliques in a given n -vertex graph. Here, ω < .
373 is the square matrixmultiplication exponent [Gal14, Wil12]. Lincoln, Vassilevska Williams, and Williams [LWW18]imposed the assumption that detecting a k -clique in an n -vertex graphs requires time n ωk/ − o (1) ω is currently not known, and, as a consequence, the precise time complexity of counting k -cliques is not well understood. Complexity of Biclique Counting.
We mention in passing some algorithmic results concernedwith finding or counting bicliques. The results below consider the case where a and b are given asinput. Binkele-Raible, Fernau, Gaspers, and Liedloff [BRFGL10] proved that, for given a, b and agraph G , one can find a K a,b subgraph in G in time O (1 . n ). Couturier and Kratsch [CK12] gavean O (1 . n )-time algorithm for Emb ( K a,b ) . They also provided an O (1 . n )-time countingalgorithm that works on bipartite graphs. It is known that the number of distinct maximal inducedbiclique subgraphs in any n -vertex graph is O (3 n/ ) = O (1 . n ) [GKL12]. If a given graph isbipartite, one can solve Emb ( K a,b ) by enumerating all maximal K a,b subgraphs using a polynomialdelay algorithm [MU04]. Kutzkov [Kut12] presented an O (1 . n )-time counting algorithm, whichis currently the fastest one. If a ≤ b are small, we can solve Emb ( K a,b ) in time O ( n a +1 ) byenumerating all size- a vertex subsets. If a = 2, we can solve Emb ( K ,b ) in time O ( n ω ) by computing A , where A ∈ { , } n × n is the adjacency matrix of a given graph. To the best of our knowledge,no other algorithms are known for Emb ( K a,b ) .Finding K a,a is NP -complete if a is given as input [GJ79]. The parameterized complexity offinding K a,a (parameterized by k ) has gathered special attention. Lin [Lin15, Lin18] proved theW[1]-hardness, which had been a fundamental open question. Moreover, his proof implies that,assuming ETH, one cannot find a K a,a in time n o ( √ a ) . However, it still remains open whether ETHrules out an n o ( a ) -time algorithm for the problem of finding a K a,a subgraph. Theorem 2.2 rulesout an n o ( a ) -time algorithm for Emb ( K a,a ) under ETH. Average-Case Complexity in P . A detection variant of (
Emb ( H )col , G ( H ) n, / ) (i.e., the problemof deciding whether a given graph G ⊆ K n × H contains H as a colorful subgraph) has beenstudied in the literature of average-case circuit complexity of the subgraph isomorphism problem(cf. Rossman [Ros18]).In a pioneering work of Ball, Rosen, Sabin, and Vasudevan [BRSV17a], they initiated the studyof average-case complexity in the context of fine-grained complexity. Ball et al. [BRSV17a] and theirsubsequent work [BRSV18] constructed average-case hard tasks by encoding worst-case problemsby a low-degree polynomial over a large finite field. Based on techniques of random self-reducibility(e.g. [CPS99]), they explored the average-case hardness of the evaluation of this polynomial underthe worst-case assumptions including the Orthogonal Vector Conjecture, APSP Conjecture, and3SUM Conjecture, recent hot conjectures in the study of hardness in P [LPW17, Wil15]. Theirwork is motivated by the construction of PoW systems. Due to the construction, their average-caseproblems are artificial.Goldreich and Rothblum [GR18a] studied the average-case complexity of Emb ( K k ) for a con-stant k . They presented a simple distribution over (cid:101) O ( n )-vertex graphs on which it is hard to countthe number of k -cliques with a success probability better than 3 /
4. The distribution is constructedby a gadget reduction, and it is somewhat artificial. The key idea of their reduction is to con-sider counting weighted cliques: The input graph has node and edge weights in F q , and the taskis to compute the sum of all weights of clique subgraphs. The weight of a clique is defined asthe product of all node weights and edge weights contained in the clique. They represented thiscounting problem as a low-degree polynomial P : F n × nq → F q and used polynomial interpolation toreduce evaluating P to computing P ( r ), where r ∼ Unif( F n × nq ). Combining the Chinese ReminderTheorem, a vertex-blowing-up technique and unifying multiple instances into one instance, they12urther reduced evaluating P ( r ) to solving Emb ( K k ) in a specific random graph. Their resulthas an error tolerance of constant probability. However, the blowing-up technique and unifyinginstances yielded an artificial random graph distribution.The proof of Theorem 2.5 is based on techniques of Boix-Adser`a, Brennan, and Bresler [BABB19],who reduced Emb ( K k ) to ( Emb ( K k ) , G n,p ), where G n,p is the distribution of an n -vertex Erd˝os-R´enyi graph with edge density p . The reduction runs in time p − n polylog n . Here, the errorprobability of the average-case solver is assumed to be at most (log n ) − C for a sufficiently largeconstant C = C ( k ). They also presented a parity variant of Emb ( K k ) and obtained a worst-case-to-average-case reduction with a better error tolerance.Independently of our work, Dalirrooyfard, Lincoln, and Vassilevska Williams [DLW20] reduced Emb ( H )col to ( Emb ( H ) , G n,p ) for a constant p . They first reduced Emb ( H )col to ( Emb ( H )col , G ( H ) n, / )by the same way as the proof of Theorem 2.5 and then reduced ( Emb ( H )col , G ( H ) n, / ) to ( Emb ( H )col , G n,p ).Using the techniques in the latter reduction (called Inclusion-Edgeclusion in their paper), we canreduce (
Emb ( H )col , G ( H ) n, / ) to ( Emb ( H ) , G n, / ). Hardness Amplification in P . The authors of [GK20] studied hardness amplification of opti-mization problems, including problems in P. Unlike our settings (in which it is highly non-trivialto construct a selector as in Theorem 2.9), it is trivial to construct a selector for any optimizationproblem; therefore, it is easy to obtain hardness amplification theorems of optimizations problemsby using the powerful direct product theorem of Impagliazzo, Jaiswal, Kabanets, and Wigder-son [IJKW10]. We present formal definitions of our framework in Section 3. In Section 4, we present the worst-case-to-average-case reduction for
Emb ( H )col and prove Theorem 2.5. In Section 5, we investigatethe worst-case complexity of Emb ( K a,b ) . In Section 6, we present the doubly-efficient interactiveproof system of Theorem 1.5. In Section 2.4, we prove the direct product theorem in the setting offine-grained complexity. Finally, in Section 8, we prove our fine-grained XOR Lemma.Here is the organization of the proofs of our main results. Theorem 1.2.
The first statement follows from Theorems 2.1 and 2.7. We can obtain Theo-rem 2.7 by combining Theorem 2.5 and Proposition 2.6. See Section 4 and Section 5.5 for theproofs of Theorem 2.5 and Proposition 2.6, respectively. The second statement is equivalent toProposition 2.3, which is shown in Section 5.4.
Theorem 1.3.
The proof is given in Section 7.2.
Theorem 1.4.
The proof is given in Section 8.2.
Theorem 1.5.
We obtain Theorem 1.5 by combining Theorem 2.8 and Lemma 5.1. Theorem 2.8and Lemma 5.1 are shown in Section 6 and Section 5.1, respectively.
Theorem 1.6.
This result follows from Theorems 2.2 and 2.7. Theorem 2.2 is shown in Sec-tion 5.6. We mention that the authors of [GK20] do not seem to be aware of [IJKW10]. Formal Definitions
Notations and Computational Model.
A graph is a pair G = ( V, E ) of two finite sets V and E ⊆ (cid:0) V (cid:1) . For simplicity, we sometimes use uv to abbreviate an edge { u, v } . We denote by V ( G )and E ( G ) the vertex and edge set of G , respectively. We sometimes identify a graph G with avector x G ∈ { , } E ( H ) by regarding x G as the edge indicator of G . For a finite set S and a positiveinteger k , let ( S ) k = { ( s , . . . , s k ) : { s , . . . , s k } ∈ (cid:0) Sk (cid:1) } .Throughout the paper, our computational model is the O (log n )-Word RAM model. As aconsequence, we assume that any field operation can be done in constant-time if the underlyingfield is F q with q = n O (1) ( n is specified by the problem). Subgraph Counting Problem.
The tensor product X × Y of two graphs X and Y is a graphgiven by V ( X × Y ) = V ( X ) × V ( Y ) and { ( x , y ) , ( x , y ) } ∈ E ( X × Y ) if and only if { x , x } ∈ E ( X )and { y , y } ∈ E ( Y ).For two graphs G and H , a mapping φ : V ( H ) → V ( G ) is homomorphism from H to G if { φ ( u ) , φ ( v ) } ∈ E ( G ) whenever { u, v } ∈ E ( H ). An embedding is an injective homomorphism. Let Emb ( H ) ( G ) be the number of embeddings from H to G .For a fixed graph H , we consider the problem Emb ( H ) of computing Emb ( H ) ( G ) for an inputgraph G . Note that Emb ( H ) is equivalent to the problem of counting subgraphs isomorphic to H :If Aut( H ) is the set of automorphisms of H and sub( H → G ) is the number of subgraphs of G that is isomorphic to H , then Emb ( H ) ( G ) = | Aut( H ) | · sub( H → G ). In this paper, we considerthe colorful variant Emb ( H )col of Emb ( H ) , defined in Section 2.2. Distributional problems, Heuristics and Reduction.
We regard a problem
Π as a functionfrom an input to the solution. An algorithm is assumed to be randomized (unless otherwise stated).An algorithm A is said to solve a problem Π in time T ( n ) if A runs in time T ( n ) for any input x of size n and Pr A [ A ( x ) = Π( x )] ≥ .A distributional problem is a pair (Π , D ) of a problem Π and a family of distributions D =( D , D , . . . ), where each D n denotes a distribution over inputs of size n . To simplify notations, weshall refer to (Π , D n ) rather than (Π , ( D n ) n ∈ N ). Throughout the paper, Π is a graph problem andeach D n is some random graph distribution.In this paper, we follow the common notion of average-case complexity (e.g., [BT06]). We saythat a (deterministic) heuristic algorithm A solves a distributional problem (Π , D ) if, for every n ∈ N , A outputs the solution of Π on input x with high probability over the random choice of x ∼ D n . The definition can be extended to a randomized (two-sided-error) heuristic algorithm: Definition 3.1 (Randomized Heuristics [BT06]) . Let (Π , D ) be a distributional problem and δ : N → [0 , be a function. We say that a randomized algorithm A solves (Π , D ) with success probability p if, for every n ∈ N , Pr x ∼D n [Pr A [ A ( x ; n ) = Π( x )] ≥ ] ≥ p . Such an algorithm A is called a (two-sided-error) randomized heuristic algorithm for (Π , D ) . The reader is referred to the survey of Bogdanov and Trevisan [BT06] for detailed background.We will use the following notion of (fine-grained) reductions.
Definition 3.2 (Average-Case-to-Average-Case Reduction) . Let (Π , D ) , (Π , D ) be distribu-tional problems. Solving a distributional problem (Π , D ) with success probability − δ is (quasi-linear-time polylog-query) reducible to solving (Π , D ) with success probability − δ if, there ex-ists a (cid:101) O ( n ) -time polylog ( n ) -query randomized oracle algorithm A such that the algorithm A solves An automorphism is a bijective homomorphism from H to H . , D ) with success probability − δ given oracle access to an arbitrary algorithm that solves (Π , D ) with success probability − δ . Definition 3.3 (Worst-Case-to-Average-Case Reduction) . Let Π be a problem, and let (Π , D ) be a distributional problem. A problem Π is (quasi-linear-time polylog-query) reducible to solving (Π , D ) with success probability − δ if, there exists a (cid:101) O ( n ) -time polylog ( n ) -query randomizedoracle algorithm A such that the algorithm A solves Π given oracle access to an arbitrary algorithmthat solves (Π , D ) with success probability − δ . EMB ( H )col In this section, we present a proof of Theorem 2.5, that is, a worst-case-to-average-case reductionfrom
Emb ( H )col to the distributional problem ( Emb ( H )col , G ( H ) n, / ).For a fixed graph H and a prime q > n | V ( H ) | , let EMBCOL n,H,q : F E ( K n × H ) q → F q be a polynomialdefined as EMBCOL n,H,q ( x ) = (cid:88) v ,...,v k ∈ V ( K n × H ) c ( v i )= i ( ∀ i ) (cid:89) { i,j }∈ E ( H ) x [ v i v j ] . (1)If x ∈ { , } E ( K n × H ) is the edge indicator of a graph G ⊆ K n × H , then EMBCOL n,H,q ( x ) = Emb ( H )col ( G ) mod q = Emb ( H )col ( G ) as q > n | V ( H ) | . For a graph H and a set S , let U ( H ) n ( S ) be theuniform distribution over S E ( K n × H ) . We sometimes identify F q with the set { , . . . , q − } . Theproof of Theorem 2.5 consists of two steps. EMBCOL n,H,q ( · ) over U ( H ) n ( F q ) First, we reduce evaluating
EMBCOL n,H,q ( x ) for a given x to solving the distributional problem( EMBCOL n,H,q ( · ) , U ( H ) n ( F q )) for a large prime q > n | V ( H ) | . Note that we can obtain such a prime q as follows. Sample a random integer r from { n | V ( H ) | , n | V ( H ) | + 1 , . . . , n | V ( H ) | } and then runthe primality test for r (according to the Prime Number Theorem, r is prime with probabilityΩ(1 / log n ).)The following is well known in the context result of random self-reducibility. A precise estimationof the running time was given by [BABB19, BRSV17a]. Lemma 4.1 (Essentially given in Lemma 3.2 of [BRSV17a]) . Let P : F Nq → F q be a multivariatepolynomial of degree d for a prime q > d . Suppose that there is a T ( N, q, d ) -time algorithm A satisfying Pr x ∼ Unif( F Nq ) [ A ( x ) = P ( x )] ≥ − δ, where δ ∈ (0 , / . Then, there is a randomized algorithm B that computes P ( y ) on input y ∈ F Nq with probability / in time O ( N d (log q ) + d + dT ( N, q, d )) .Proof sketch. Ball et al. [BRSV17a] proved this result under the condition that d >
9. Boix-Adser`a,Brennan, and Bresler [BABB19] obtained the same result for a prime power q > d (under thesame condition) by the same way. The common idea is to invoke the well-known local decoding ofthe Reed-Muller code (see, e.g., [Lip91, GS92]). In this paper, we just modify a parameter appeared15n their proof to remove the degree condition. We briefly describe the algorithm and refer to thefull version of [BRSV17a] for the analysis.For a given y ∈ F Nq , sample two random vectors z , z ∼ Unif( F Nq ) independently, and con-sider the univariate function f ( t ) := y + z t + z t . Note that our task is to compute f (0). Set m = 100 d (the authors of [BRSV17a] set m = 12 d ). Use the oracle algorithm A and compute A ( f (1)) , . . . , A ( f ( m )). By the Berlekamp–Welch decoding [BW86], obtain a polynomial ˆ f andoutput ˆ f (0).By applying this result to our setting, we obtain the following. Corollary 4.2.
For a fixed graph H and a prime n | V ( H ) | < q < n | V ( H ) | , let EMBCOL n,H,q ( · ) bethe polynomial given in (1) . Suppose that there is a T ( n ) -time algorithm A satisfying Pr x ∼U ( H ) n ( F q ) [ A ( x ) = EMBCOL n,H,q ( x )] ≥ . Then, there is a randomized algorithm B that computes EMBCOL n,H,q ( y ) on input y ∈ F E ( K n × H ) q with success probability / in time O ( n (log n ) + T ( n )) . EMBCOL n,H,q ( U ( H ) n ( F q )) to EMBCOL n,H,q ( U ( H ) n ( { , } )) We reduce the problem of computing
EMBCOL n,H,q ( · ) over the distribution U ( H ) n ( F q ) to that over U ( H ) n ( { , } ) based on the binary extension technique of [BABB19]. Observe that the distributionalproblem ( EMBCOL n,H,q ( · ) , U ( H ) n ( { , } )) is equivalent to ( Emb ( H )col , G ( H ) n, / ) if q > n | V ( H ) | . Lemma 4.3.
Let H be a fixed graph and q be a prime satisfying n | V ( H ) | < q < n | V ( H ) | . Supposethere is a T ( n ) -time randomized heuristic algorithm A satisfying Pr x ∼U ( H ) n ( { , } )) (cid:20) Pr A [ A ( x ) = EMBCOL n,H,q ( x )] ≥ (cid:21) ≥ − δ, where δ = (log n ) − C for a sufficiently large constant C = C H > that depends on H .Then, there is a T ( n ) · polylog n -time randomized heuristic algorithm B satisfying Pr x ∼U ( H ) n ( F q ) (cid:20) Pr B [ B ( x ) = EMBCOL n,H,q ( x )] > (cid:21) > . Note that Theorem 2.5 follows from Corollary 4.2 and Lemma 4.3.
Observation.
Suppose that, for each uv ∈ E ( K n × H ), x [ uv ] ∈ F q can be rewritten as x [ uv ] = t − (cid:88) l =0 l · z ( l ) [ uv ] mod q (2)16or some binary variables z (0) [ uv ] , . . . , z ( t − [ uv ] ∈ { , } . Here, t is some large integer that will bespecified later. Then, we obtain EMBCOL n,H,q ( x ) = (cid:88) v ,...,v k ∈ V ( G ) c ( v i )= i ( ∀ i ) (cid:89) ij ∈ E ( H ) t − (cid:88) l =0 l · z ( l ) [ v i v j ]= (cid:88) v ,...,v k ∈ V ( G ) c ( v i )= i ( ∀ i ) (cid:88) a ∈{ ,...,t − } E ( H ) (cid:89) ij ∈ E ( H ) a [ ij ] · z ( a [ ij ]) [ v i v j ]= (cid:88) a ∈{ ,...,t − } E ( H ) (cid:80) e ∈ E ( H ) a [ e ] (cid:88) v ,...,v k ∈ V ( G ) c ( v i ) ∈ i ( ∀ i ) (cid:89) ij ∈ E ( H ) z ( a [ ij ]) [ v i v j ] . = (cid:88) a ∈{ ,...,t − } E ( H ) (cid:80) e ∈ E ( H ) a [ e ] · EMBCOL n,H,q ( χ ( a ) ) . (3)Here, we define χ ( a ) [ uv ] := z ( a [ c ( u ) c ( v )]) [ uv ] ∈ { , } for each uv ∈ E ( K n × H ).Thus, our goal is to sample z such that the distribution of z ( a ) is closed to G ( H ) n, / for each a ∈ { , . . . , t − } E ( H ) . In this paper, we invoke a special case of Lemma 4.3 of [BABB19] andimprove the running time of a sampling procedure. Lemma 4.4.
Let q > be a prime and t be some integer. For each x ∈ F q , let M x := { m ∈{ , . . . , t − } : m mod q = x } and Y x ∼ Unif( M x ) be a random variable. Let Y R be the distributionof Y R for R ∼ Unif( F q ) . Then, the following hold.1. d TV ( Y R , Unif( { , . . . , t − } )) ≤ Cq/ t for some absolute constant C .2. For any given x ∈ F q , we can sample Y x in time O ( t ) . Corollary 4.5.
Let t be some integer. Let Z , . . . , Z t − ∼ Unif( { , } ) be i.i.d. random variables.Then, for any given x ∈ F q , we can sample t random variables z , . . . , z t − satisfying the followingin time O ( t ) .1. It holds that (cid:80) t − i =0 i · z i mod q = x .2. The distribution of ( z , . . . , z t − ) when x is sampled from Unif( F q ) is of total variation distanceat most O ( q/ t ) from the uniform distribution ( Z , . . . , Z t − ) .Proof. For a given x ∈ F q , let z , . . . , z t − be the binary expansion of Y x of Lemma 4.4. Then, Y x = (cid:80) t − i =0 i · z i = x (mod q ) by the definition of Y x . Let Y := (cid:80) t − i =0 i · Z i ∼ Unif( { , . . . , t − } ).Let f : { , . . . , t − } → { , } t denote the function that maps y ∈ { , . . . , t − } to the binaryrepresentation of y . Note that f is a bijection and f ( Y x ) = ( z , . . . , z t − ) holds. Then, fromLemma 4.4, for any A ⊆ { , } t , we have | Pr[( z , . . . , z t − ) ∈ A ] − Pr[( Z , . . . , Z t − ) ∈ A ] | = | Pr[ Y x ∈ f − ( A )] − Pr[ Y ∈ f − ( A )] | = O ( q/ t ) . This implies the statement 2 of Corollary 4.5.
Remark . Boix-Adser`a, Brennan, and Bresler [BABB19] considered the general case of Z i ∼ Ber( c i ), where Ber( c i ) is the Bernouli random variable with success probability c i . Roughly speak-ing, for some t = Θ( c − (1 − c ) − log( q/(cid:15) ) log q ), they proved (1) d TV ( L ( Y ) , L ( Y R )) ≤ (cid:15) , and (2)17or any given x ∈ F q , Y x can be sampled in time O ( tq ). Since q > n V ( H ) , the sampling of Y x cannot be applied directly due to the running time O ( tq ). To avoid the large running time, Boix-Adser`a, Brennan, and Bresler [BABB19] used the Chinese Reminder Theorem to reduce computing EMBCOL n,H,q ( · ) to the computing EMBCOL n,H,q ( · ) , . . . , EMBCOL n,H,q m ( · ), where q , . . . , q m aresmall primes. In Lemma 4.4, we focus on the special case of c i = 1 / Y x .We will present the proof of Lemma 4.4 later. Proof of Lemma 4.3.
We describe the randomized algorithm B that computes EMBCOL n,H,q ( x )for a given x ∼ U ( H ) n ( F q ).Set t = K log q for a sufficiently large constant K = K ( H ) that will be chosen later dependingonly on H . For each e ∈ E ( K n × H ), do the following: For x = x [ e ] ∈ F q , sample z [ e ] =( z [ e ] , . . . , z t − [ e ]) of Corollary 4.5 in time O ( t ). Note that (2) holds.After sampling ( z [ e ]) e ∈ E ( K n × H ) , the algorithm B computes EMBCOL n,H,q ( x ) using (3): For each a ∈ { , . . . , t − } E ( H ) , construct χ ( a ) using ( z [ e ]) e ∈ E ( K n × H ) and compute EMBCOL n,H,q ( χ ( a ) ) usingthe T ( n, H )-time heuristic algorithm A that solves ( EMBCOL n,H,q ( · ) , U ( H ) n ( { , } )) with successprobability 1 − δ .We claim that B has success probability 1 − t | E ( H ) | δ − O ( n | E ( H ) | q/ t ), which completes theproof of Lemma 4.3: Indeed, choosing t = K log n for a sufficiently large constant K = K ( H ), thesuccess probability of B is at least 1 − O ( δ (log n ) | E ( H ) | ) − o (1) ≥ / δ = o ((log n ) − | E ( H ) | ). Success probability of B . Since x [ e ] ∼ Unif( F q ), Lemma 4.4 implies that the distributionof z [ e ] := ( z i [ e ]) i ∈{ ,...,t − } is total variation distance at most (cid:15) := O ( q/ t ) from that of Z [ e ] :=( Z [ e ] , . . . , Z t − [ e ]), where Z [ e ] , . . . , Z t − [ e ] ∼ Unif( { , } ) are i.i.d. random variables. Therefore,the distribution of z = ( z [ e ]) e ∈ E ( K n × H ) is total variation distance at most | E ( K n × H ) | (cid:15) from Z = ( Z [ e ]) e ∈ E ( K n × H ) (here, z [ e ] are independent as well as Z [ e ]).Let A be the randomized heuristic algorithm described in Lemma 4.3. Let S be the set ofgraphs that is solved by A . Formally, S = (cid:26) F ⊆ K n × H : Pr A [ A ( F ) = Emb ( H )col ( F )] ≥ (cid:27) . Let z := ( z [ e ]) e ∈ E ( K n × H ) and Z := ( Z [ e ]) e ∈ E ( K n × H ) be random variables described above. Foreach a ∈ { , . . . , t − } E ( H ) , we have Pr Z (cid:2) ˜ χ ( a ) ∈ S (cid:3) ≥ − δ , where ˜ χ ( a ) = ( ˜ χ ( a ) [ e ]) e ∈ E ( K n × H ) is de-fined as ˜ χ ( a ) [ uv ] := Z ( a [ c ( u ) c ( v )]) [ uv ]. Here, we identify a graph with a binary vector in { , } E ( K n × H ) .Recall that c : V ( K n × H ) → V ( H ) maps a vertex to its color. Note that the distribution of ˜ χ ( a ) is the same as G ( H ) n, / for every fixed a ∈ { , . . . , t − } E ( H ) . By the union bound, we havePr Z (cid:104) ∀ a ∈ { , . . . , t − } E ( H ) : ˜ χ ( a ) ∈ S (cid:105) ≥ − t | E ( H ) | δ. Since z is total variation distance at most | E ( K n × H ) | (cid:15) from Z , this impliesPr z (cid:104) ∀ a ∈ { , . . . , t − } E ( H ) : χ ( a ) ∈ S (cid:105) ≥ − t | E ( H ) | δ − | E ( K n × H ) | (cid:15). This completes the proof of the claim. 18 roof of Lemma 4.4.
Indeed, the statement 1 is a special case of Lemma 4.3 in [BABB19] andthe proof is already given (see p. 23 of [BABB19]). For completeness, we present the proof byfocusing on the special case. Consider the size of M x . Let N := 2 t /q . Since x ∈ { , . . . , q − } , itholds that N − ≤ (cid:22) t q − (cid:23) ≤ | M x | ≤ (cid:22) t q (cid:23) ≤ N. Let Y ∼ Unif( { , . . . , t − } ) and Y R ∼ Y R be random variables, where R ∼ Unif( F q ). For any A ⊆ { , . . . , t − } , consider the events that Y ∈ A and Y R ∈ A . ObservePr[ Y R ∈ A ] = (cid:88) x ∈ F q Pr[ Y x ∈ A ∩ M x | R = x ] Pr[ R = x ] = 1 q (cid:88) x ∈ F q | A ∩ M x || M x | and Pr[ Y ∈ A ] = | A | t = 1 q (cid:88) x ∈ F q | A ∩ M x | N .
Therefore, it holds for any A ⊆ { , . . . , t − } that | Pr[ Y R ∈ A ] − Pr[ Y ∈ A ] | ≤ q (cid:88) x ∈ F q | A ∩ M x | (cid:12)(cid:12) | M x | − − N − (cid:12)(cid:12) ≤ | A | q (cid:18) N − − N (cid:19) = | A | q · O ( N − ) ≤ O ( q/ t ) . This completes the proof of the statement 1.We show the statement 2. The sampling can be done by the following scheme: For a given x ∈ F q , let M := (cid:98) (2 t − x − /q (cid:99) = | M x | − K ∼ Unif( { , . . . , M } ). Then, output L := Kq + x . For any k ∈ { , . . . , M } ,Pr[ L = kq + x ] = Pr[ K = k ] = 1 M + 1 . In other words, L ∼ Unif( M x ) for any x . EMB ( K a,b ) This section is devoted to prove Theorems 2.1, 2.2 and 2.7 and Proposition 2.3. In Sections 5.1to 5.5, we provide several technical results. Finally, in Section 5.6, we combine these results toshow Theorems 2.1, 2.2 and 2.7 and Proposition 2.3.
We first prove that the K a,b -subgraph counting and colorful K a,b -subgraph counting are equivalent. Lemma 5.1.
Consider
Emb ( K a,b )col and Emb ( K a,b ) . Given oracle access to one of them, we cansolve the other one in time O ( a + b ) + O ( n ) (in the worst-case sense). Emb ( H )col is reducible to Emb ( H ) by using theinclusion-exclusion principle [CM14, Cur18]. Folklore 5.2.
Let H be a graph. If Emb ( H ) for n -vertex graphs can be solved in time T ( n ) , then Emb ( H )col can be solved in time O (2 | V ( H ) | T ( n )) . Now we discuss the converse direction: Can we solve
Emb ( H ) given oracle access to Emb ( H )col ?We show that Emb ( H ) is reducible to Emb ( H )col when H = K a,b . To this end, we consider theproblem Hom ( H ) that asks the number Hom ( H ) ( G ) of homomorphisms from H to a given graph G . Recall that a mapping φ : V ( H ) → V ( G ) is a homomorphism if { φ ( u ) , φ ( v ) } ∈ E ( G ) whenever { u, v } ∈ E ( H ).We reduce Emb ( K a,b ) to Emb ( K a,b )col by the following three steps. First, we show that Hom ( H ) ( G ) is equal to Emb ( H )col ( G × H ) (Fact 5.3). Second, we use Lov´asz’s identity [Lov12] toreduce Emb ( H ) to Hom ( H (cid:48) ) for some family of graphs H (cid:48) (Theorem 5.4). Finally, we observethat Hom ( H (cid:48) ) is reducible to Emb ( K a,b ) when H = K a,b (Proposition 5.5).The following well-known fact asserts that Hom ( H ) is reducible to Emb ( H )col . Fact 5.3.
Let H be a fixed k -vertex graph. For any graph G , it holds that Hom ( H ) ( G ) = Emb ( H )col ( G × H ) . Consequently, if Emb ( H )col can be solved in time T ( kn ) on kn -vertex graphs,then Hom ( H ) can be solved in time O ( T ( kn ) + kn ) on n -vertex graphs.Proof. We can solve
Hom ( H ) on input G as follows. Construct G × H and then run the algorithmfor Emb ( H )col on input G × H . Now we show Hom ( H ) ( G ) = Emb ( H )col ( G × H ). Let φ be ahomomorphism from H to G . Then, the mapping ψ : V ( H ) (cid:51) v (cid:55)→ ( φ ( v ) , v ) ∈ V ( G × H ) is alsoa homomorphism and moreover it is injective. This correspondence between φ and ψ is one-to-one.In light of Fact 5.3, it suffices to reduce Emb ( H ) to Hom ( H ) . To this end, we invoke thefollowing identity. Theorem 5.4 (Lov´asz [Lov12]; See (2) of [CDM17]) . Let H be a fixed graph. Let P ( H ) be theset of partitions of V ( H ) such that, for every π = { B , . . . , B t } ∈ P ( H ) , each B i ⊆ V ( H ) is anindependent set ( i = 1 , . . . , t ). For each π ∈ P ( H ), define H/π as the graph obtained by contractingeach vertex set in π . Then Emb ( H ) ( G ) = (cid:88) π ∈P ( H ) ( − | V ( H ) |−| π | (cid:89) B ∈ π ( | B | − · Hom ( H/π ) ( G ) . Here, | π | denotes the number of subsets in π .Combining Theorem 5.4 and Fact 5.3, we can reduce Emb ( H ) to solving a family of problems( Emb ( H/π )col ) π ∈P ( H ) . If H = K a,b , we can enumerate all elements of P ( H ) in time O (2 a + b ), and thusthe reduction runs in time O ( n + 2 a + b ). Moreover, we show in Proposition 5.5 that Emb ( K a,b /π )col is reducible to Emb ( K a,b )col for every π ∈ P ( K a,b ), which enables us to reduce Emb ( K a,b ) to Emb ( K a,b )col . Proposition 5.5.
Assume that
Emb ( K a,b )col can be solved in time T ( n ) . Let π ∈ P ( K a,b ) . Then, Emb ( K a,b /π )col can be solved in time O ( n + T ( n )) . roof. Observe that, for any π ∈ P ( K a,b ), we have K a,b /π = K c,d for some constants c ≤ a and d ≤ b ; therefore, it suffices to reduce Emb ( K c,d )col to Emb ( K a,b )col .Let ( n, G ) be an input of Emb ( K c,d )col , where G ⊆ K c,d × K n . Regard the vertices in V ( K a,b )as V ( K a,b ) = { l , . . . , l a , r , . . . , r b } so that E ( K a,b ) = { l i , r j } i ∈ [ a ] ,j ∈ [ b ] . Then, each vertex v ∈ V ( G )can be represented as the form ( r i , u ) or ( l i , u ). We write V ( G ) = R ∪ L , where R is the setof vertices of the form ( r i , u ), and L is that of the form ( l i , u ). Fix a vertex v ∈ V ( K n ) and let L add = { ( l i , v ) } ai = c +1 and R add = { ( r i , v ) } bi = d +1 be vertex sets. We construct a graph ˆ G ⊆ K a,b × K n as follows. V ( ˆ G ) = V ( G ) ∪ L add ∪ R add ,E ( ˆ G ) = E ( G ) ∪ E ( R add , L ∪ L add ) ∪ E ( L add , R ∪ R add ) , where, for two vertex subsets S and T , E ( S, T ) = { s,t } s ∈ S,t ∈ T . See Figure 3 for an illustration. RL R add L add Figure 3: The graph ˆ G of the reduction. In this figure, Emb ( K , )col ( G ) = Emb ( K , )col ( ˆ G ) holds.Note that Emb ( K a,b )col ( ˆ G ) = Emb ( K c,d )col ( G ) holds since there is a one-to-one correspondencebetween copies of K a,b in ˆ G and that of K c,d in G .Lemma 5.1 follows from Folklore 5.2, Fact 5.3, Theorem 5.4, and Proposition 5.5. Remark . We comment on the relationship between
Hom ( H ) and Emb ( H ) . It is easy to seethat the problems Hom ( K k ) and Emb ( K k ) are equivalent. More generally, such an equivalenceholds if H is a core ; here, a graph H is said to be a core if any homomorphism from H to H isan isomorphism. However, for some H , it is widely believed that there is a gap between Hom ( H ) and Emb ( H ) : For example, let M k be the graph of disjoint k edges. It is known that Emb ( M k ) (which is the problem of counting the number of matchings of size k ) is Hom ( M k ) can be solved in linear time (observe that Emb ( M k ) ( G ) = (2 | E ( G ) | ) k ). K a,b -DETECTION Assume a ≤ b . By enumerating all subsets of size a , we can solve both Colorful K a,b -Detection in time O ( n a +1 ). If a given graph G is sparse and has m edges, we can solve the problem in time O ( m a ) by enumerating (cid:0) N ( v ) a (cid:1) for every vertex v , where N ( v ) denotes the set of vertices adjacentto v . 21 heorem 5.7 (Reminder of Theorem 2.4) . For any constants a ≥ and (cid:15) > , there exists aconstant b = b ( a, (cid:15) ) ≥ a such that Colorful K a,b -Detection cannot be solved in time O ( m a − (cid:15) ) unless SETH fails, where m is the number of edges of the input graph.Remark . Theorem 2.1 immediately follows from Folklore 5.2 and Theorem 2.4.In the proof of Theorem 2.4, we consider k -Orthogonal Vectors ( k - OV ). In k - OV , we aregiven sets A , . . . , A k ⊆ { , } d of binary vectors each of cardinarity n and dimension d satisfying d ≤ K log n for a constant K . Our task is to decide whether there exist vectors a , . . . , a k such that a i ∈ A i for any i and (cid:80) dj =1 (cid:81) ki =1 a i [ j ] = 0. The na¨ıve exhaustive search solves k -OV in time O ( n k d ) = O ( n k log n ). The current known fastest algorithm solves it in time O ( n k − /O (log( d/ log n )) ) [AWY15].The k -Orthogonal Vectors Conjecture ( k -OVC) asserts that k - OV requires time n k − o (1) for any d = ω (log n ): More precisely, under k -OVC, for any k ≥ (cid:15) >
0, there exists a constant K ≥ O ( n k − (cid:15) )-time algorithm solves k - OV of dimension d ≤ K log n . It is known that, forevery constant k ≥
2, SETH implies k -OVC [Wil15, Wil05, LPW17]. Thus, it suffices to reduce k - OV to Emb ( K a,b ) for k = a . The reduction (Proof of Theorem 2.4).
Fix any constant a ≥
2. Assume that there exists aconstant (cid:15) >
Colorful K a,b -Detection can be solved in O ( m a − (cid:15) ) for every b ≥ a .We will prove that, under this assumption, there exists a constant (cid:15) (cid:48) > a - OV ofdimension d = K log n can be solved in time O ( m a − (cid:15) (cid:48) ) for any K . To this end, we present a many-to-one reduction: The reduction maps an instance of a - OV to an equivalent instance of Colorful K a,b -Detection .Let (cid:15) > A , . . . , A k ⊆ { , } d be an instance of k -OV of dimension d = K log n . We identify a vector x ∈ { , } d with a subset x ⊆ [ d ]. Thus, each A i is identified with a family of subsets of [ d ]. Let P ∪ · · · ∪ P C be the partitionof [ d ] such that |P i | ≤ (cid:15) log n holds for every i ∈ [ C ], where C = K/(cid:15) (we will choose (cid:15) so that
K/(cid:15) is an integer).The reduction constructs a graph G and a coloring c : V ( G ) → [ a + b ], resulting in an instanceof Colorful K a,b -Detection of a := k and b := C . The vertex set of G is of the form V ( G ) = V ∪ · · · ∪ V k ∪ W ∪ · · · ∪ W C , where each subset V , . . . , V k , W , . . . , w C is assigned with a distinct color. For each subset a ∈ A i ,we create a vertex v a ∈ V i . For each index j ∈ [ C ], enumerate all subsets of P j . We associate a k -tuple z = ( y , . . . , y k ) ∈ (2 P j ) k of the subsets with a vertex w z ∈ W j , if the corresponding vectors y , . . . , y k are orthogonal on P j . Formally, the vertex set V ( G ) is V i := { v a : a ∈ A i } ,W j := w z : z = ( y , . . . , y k ) ∈ (2 P j ) k satisfies (cid:88) r ∈P j (cid:89) s ∈ [ k ] y s [ r ] = 0 . Two vertices v a ∈ V i and w z ∈ W j of z = ( y , . . . , y k ) are joined by an edge if a ∩ P j = y i ∩P j holds. The edge set E ( G ) contains no other edges. Note that G ⊆ K n × K a,b in which V , . . . , V a , W , . . . , W b obtain distinct colors. Correctness.
Let A , . . . , A k ⊆ { , } [ d ] be the instance of k -OV with d = K log n and G be thegraph constructed by the reduction above. Recall that each A i is identified with a family of n subsets22f [ d ]. Suppose that the given instance is a YES-instance. Then, there is a k -tuple ( a , . . . , a k ) ∈ A × · · · × A k such that the corresponding vectors a , . . . , a k satisfy (cid:80) r ∈ [ d ] (cid:81) s ∈ [ k ] a s [ r ] = 0. Let U := (cid:91) i ∈ [ k ] { v a i } ⊆ V ( G ) ,W := (cid:91) j ∈ [ C ] { w z ∈ W j : z = ( y , . . . , y k ) where each y i satisfies y i ∩ P j = a i ∩ P j } . The set U ∪ W induces a colorful subgraph isomorphic to K a,b , where a = k and b = C ; thus, thepair ( G, c ) of a graph G and coloring c is a YES-instance of Colorful K a,b -Detection .Conversely, suppose that G contains a colorful subgraph H isomorphic to K a,b . Then we have | V ( H ) ∩ V i | = | V ( H ) ∩ W j | = 1 for every i ∈ [ k ] and j ∈ [ C ]. Let v i ∈ V ( H ) ∩ V i and w j ∈ V ( H ) ∩ W j .As H is isomorphic to K a,b , { v i , w j } ∈ E ( G ) for every i, j . Let a i ∈ { , } d be the vector associatedwith the vertex v i . For every j ∈ [ C ], we have (cid:80) r ∈P j (cid:81) s ∈ [ k ] a s [ r ] = 0 since each w j is incident to v i for all i ∈ [ k ]. Thus, we have (cid:80) r ∈ [ n ] (cid:81) s ∈ [ k ] a s [ r ] = 0 and hence ( A , . . . , A k ) is a YES-instanceof k -OV. Running time.
The size of the constructed graph G satisfies | V ( G ) | ≤ kn + Cn (cid:15)k , | E ( G ) | ≤ kCn (cid:15)k . Thus, if
Colorful K a,b -Detection on G can be solved in time O ( m a − (cid:15) (cid:48) ), letting (cid:15) > (cid:15)k )( k − (cid:15) (cid:48) ) ≤ k − (cid:15) (cid:48) / O ( m a − (cid:15) (cid:48) ) = O ( n (1+ (cid:15)k )( k − (cid:15) (cid:48) ) ) = O ( n k − (cid:15) (cid:48) / )time algorithm for k - OV . This falsifies k -OVC as well as SETH. K a,a -DETECTION Consider the decision problem K a -Detection in which we are asked to decide whether the givengraph contains a clique of size a or not. In this section, we reduce K a -Detection to Colorful K a,a -Detection . Note that K a -Detection does not admit an f ( k ) · n o ( k ) -time algorithm forany function f ( · ) unless ETH fails [CHKX06]; thus, the reduction establishes the ETH-hardness of Colorful K a,a -Detection . Lemma 5.9.
There is an O ( n ) -time algorithm that, given a graph G of n vertices, outputs a graph G (cid:48) ⊆ K n × K a,a of O ( an ) vertices such that G contains an a -clique if and only if G (cid:48) contains acolorful K a,a -subgraph.Proof. Let G be an instance of K a -Detection . We transform G to the graph G (cid:48) mentioned inLemma 5.9.Let U , . . . , U a , W , . . . , W a be copies of V ( G ). For notational convenience, we write V ( G ) = { v , . . . , v n } and U i = { u ( i )1 , . . . , u ( i ) n } , W i = { w ( i )1 , . . . , w ( i ) n } . Here, each u ( i ) j corresponds to v j (andso does w ( i ) j ). We set V ( G (cid:48) ) = (cid:83) i ∈ [ a ] ( U i ∪ W i ); each V i and W i is assigned with a distinct color.(More formally, each vertex in V i is assigned with a color i and each vertex in W i is assigned witha color a + i ). We construct E ( G (cid:48) ) such that, for all i, k ∈ [ a ] and j, l ∈ [ n ], an edge { u ( i ) j , u ( k ) l } is23n E ( G (cid:48) ) if either (1) i = j and j = l , or (2) i (cid:54) = j and { v j , v l } ∈ E ( G ) holds. The set E ( G (cid:48) ) doesnot contain any other edges. This graph can be constructed in time O ( an ).Now we check the correctness. Suppose that a vertex set S = { v i , . . . , v i a } forms an a -cliquein G . Then, the vertex set { u (1) i , . . . , u ( a ) i a , w (1) i , . . . , w ( a ) i a } forms a colorful K a,a -subgraph in G (cid:48) .Conversely, if the set { u (1) i , . . . , u ( a ) i a , w (1) j , . . . , w ( a ) j a } forms a colorful K a,a -subgraph in G (cid:48) , then itholds that i = j , . . . , i a = j a and the set { v i , . . . , v i a } forms an a -clique in G . n a + o (1) -Time Algorithm for EMB ( K a,b ) We now present an algorithm that matches the lower bounds presented so far. Specifically, wedesign an algorithm that solves
Emb ( K a,b )col in time O ( bn a + o (1) ), thereby proving Proposition 2.3.The algorithm of Proposition 2.3 is similar to the O ( n k + o (1) )-time algorithm for k -DominatingSet of k ≥ n × n γ matrix and an n γ × n matrix in n o (1) arithmetic operations if γ ≤ . G = ( V, E ) be a given instance of
Emb ( K a,b ) . We first consider the case when a is even.We construct an (cid:0) na/ (cid:1) × n matrix B as follows: For each S ∈ (cid:0) Va/ (cid:1) and v ∈ V , B [ S ][ v ] = (cid:40) S ⊆ N ( v ) , . Then, compute the product BB (cid:62) by the fast rectangular matrix multiplication [GU18]. Therunning time is O ( n a + o (1) ) if a ≥
8. Notice that BB (cid:62) [ S ][ S ] is equal to the size of the vertexsubset W ( S , S ), where W ( S , S ) := { v ∈ V \ ( S ∪ S ) : S ∪ S ⊆ N ( v ) } . In other words, theset W ( S , S ) contains vertices that is adjacent to all vertices in S ∪ S . For any S , S ∈ (cid:0) Va/ (cid:1) with S ∩ S = ∅ and T ∈ (cid:0) W ( S ,S ) b (cid:1) , the vertex set S ∪ S ∪ T forms a K a,b subgraph. On theother hand, for a K a,b subgraph, there are c (cid:0) aa/ (cid:1) ways to take S , S , T , where c = 2 if a < b and c = 4 if a = b . If a < b , the factor c reflects the symmetry of S and S ; thus c = 2. If a = b , wefurther take the symmetry of S ∪ S and T into account; thus c = 4. Then, the number of K a,b subgraphs contained in G is given by c − (cid:18) aa/ (cid:19) − · (cid:88) S ,S ∈ ( Va/ ) : S ∩ S = ∅ (cid:18) BB (cid:62) [ S ][ S ] b (cid:19) . Now consider the case when a is odd. Fix a vertex u ∈ V . Again, we construct an (cid:0) n ( a − / (cid:1) × n matrix B ( u ) as follows: For each S ∈ (cid:0) V ( a − / (cid:1) and v ∈ V , B ( u ) [ S ][ v ] = (cid:40) { u, v } ∈ E, v (cid:54)∈ S and S ⊆ N ( v ) , . Then compute B ( u ) ( B ( u ) ) (cid:62) for all u ∈ V . Note that the multiplication can be computed in time n a − o (1) for each u ∈ V . Observe that B ( u ) ( B ( u ) ) (cid:62) [ S ][ S ] is the number of vertices that isadjacent to all vertices in S ∪ S ∪ { u } . Thus, the number of K a,b contained in G is given by c − (cid:18) a (cid:18) a − a − / (cid:19)(cid:19) − · (cid:88) u ∈ V (cid:88) S ,S ∈ ( V ( a − / ) : S ∩ S = ∅ (cid:18) B ( u ) ( B ( u ) ) (cid:62) [ S ][ S ] b (cid:19) , c = 2 if a < b and c = 4 if a = b .This yields an O ( bn a + o (1) ) time algorithm (note that (cid:0) nk (cid:1) can be computed in O ( k log n ) time). ( EMB ( K a,b )col , G ( K a,b ) n, / ) to ( EMB ( K a,b ) , K a,b,n ) In this section, we present a proof of Proposition 2.6, i.e., an average-case-to-average-case re-duction from (
Emb ( K a,b )col , G ( K a,b ) n, / ) to ( Emb ( K a,b ) , K a,b,n ). This will complete a proof of Theo-rem 2.7: Recall that Theorem 2.5 reduces Emb ( K a,b )col to ( Emb ( K a,b )col , G ( K a,b ) n, / ). Combined withFolklore 5.2, one can reduce Emb ( K a,b ) to ( Emb ( K a,b )col , G ( K a,b ) n, / ). Overall, we obtain a reductionfrom Emb ( K a,b ) to ( Emb ( K a,b ) , K a,b,n ) as stated in Theorem 2.7. Proof of Proposition 2.6.
Let B n,m, / be the distribution of a random bipartite graph with left andright vertex sets of size n and m , respectively. Let G be an input of ( Emb ( K a,b )col , G ( K a,b ) n, / ). Observethat the distribution G ( K a,b ) n, / is identical to B an,bn, / . We say that a subgraph F ⊆ G contains color i if F contains a vertex of color i . Let S be the set of subgraphs F ⊆ G isomorphic to K a,b . Let S i ⊆ S be the set of subgraphs F ∈ S that contain color i . Observe that Emb ( K a,b )col ( G ) = (cid:12)(cid:12)(cid:12)(cid:84) i ∈ V ( K a,b ) S i (cid:12)(cid:12)(cid:12) .By the inclusion-exclusion principle, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:92) u ∈ V ( K a,b ) S u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | S | − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:91) i ∈ V ( K a,b ) S i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | S | − (cid:88) ∅(cid:54) = J ⊆ V ( K a,b ) ( − | J |− (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:92) j ∈ J S j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . In light of this equality, it suffices to compute | S | and (cid:12)(cid:12)(cid:12)(cid:84) j ∈ J S j (cid:12)(cid:12)(cid:12) for all nonempty J ⊆ V ( K a,b ).Note that the set ∩ j ∈ J S j is equal to the set of K a,b subgraphs in G that does not contain any colorsfrom J . To state it more formally, for a nonempty set J ⊆ V ( K a,b ), let V J = { x ∈ V ( G ) : c ( x ) ∈ J } and G J = G [ V J ] be the induced subgraph of G by V J . Then, (cid:84) j ∈ J S j is equal to the set of K a,b subgraphs contained in G J . Suppose that we have a T ( n )-time randomized algorithm A that solves ( Emb ( K a,b ) , K a,b,n ) with failure probability δ . Note that, for each J ⊆ V ( K a,b ), thedistribution of G J for G ∼ B an,bn, / is identical to B cn,dn, / for some c ≤ a and d ≤ b ; thus, wecan obtain (cid:12)(cid:12)(cid:12)(cid:84) j ∈ J S j (cid:12)(cid:12)(cid:12) with probability at least 1 − abδ since A ( G J ) = c ! d ! (cid:12)(cid:12)(cid:12)(cid:84) j ∈ J S j (cid:12)(cid:12)(cid:12) (here, c ! d ! isthe number of automorphisms of K c,d ). Therefore, from the union bound, we can obtain (cid:12)(cid:12)(cid:12)(cid:84) j ∈ J S j (cid:12)(cid:12)(cid:12) for all ∅ (cid:54) = J ⊆ V ( K a,b ) with probability at least 1 − ab a + b δ . Moreover, A ( G ) = a ! b ! | S | holdswith probability 1 − abδ . Hence, we can solve ( Emb ( K a,b ) , G ( K a,b ) n, / ) in time O ( ab a + b · T ( n )) withprobability 1 − O ( ab a + b δ ). Theorem 2.7 follows from Folklore 5.2, Theorem 2.5, and Proposition 2.6. We can show Theorem 2.1by combining Theorem 2.4 and Lemma 5.1 (Note that we can solve
Colorful K a,b -Detection using a solver for Emb ( K a,b ) ). Similarly, Theorem 2.2 follows from Lemmas 5.1 and 5.9 and thewell-known fact that ETH rules out an n o ( a ) -time algorithm for K a -Detection [CHKX06].25 Doubly-Efficient Interactive Proof System
This section is devoted to proving Theorem 2.8. Recall that, in
Emb ( H )col , we are given n and G ⊆ K n × H and then are asked the number of colorful subgraphs F ⊆ G that is isomorphic toa fixed graph H . Fix a prime n | V ( H ) | < q < n | V ( H ) | and consider the polynomial EMBCOL n,H,q : F E ( K n × H ) q → F q defined in (1). In our interactive proof system IP , the verifier checks the statementthat EMBCOL n,H,q ( x ) = C for given C ∈ F q and x ∈ F E ( K n × H ) q . Recall that, if x is an edgeindicator vector of a graph G , then EMBCOL n,H,q ( x ) = Emb ( H ) ( G ) holds. We may assume without loss of generality that n = 2 t for some t ∈ N . (If not, we add isolatedvertices to G .) For each i ∈ V ( H ), let V i := { v ∈ V ( K n × H ) : c ( v ) = i } .For each i ∈ V ( H ), let ( V i, , V i, ) be a partition of V i such that | V i, | = | V i, | = | V i | /
2. For η ∈ { , } V ( H ) , let E η = ∪ ij ∈ E ( H ) E ( V i,η ( i ) , V j,η ( j ) ), where E ( S, T ) = { e ∈ E ( K n × H ) : e ∩ S (cid:54) = ∅ and e ∩ T (cid:54) = ∅} for S, T ⊆ V ( K n × H ) (see Figure 4). Since | V i,η ( i ) | = | V i | /
2, we can identify E η with E ( K n/ × H ). From the definition (1), we have EMBCOL n,H,q ( x ) = (cid:88) η ∈{ , } V ( H ) EMBCOL n/ ,H,q ( x [ E η ]) , (4)where x [ E η ] ∈ F E η q is the restriction of x on E η . H V V V V V V V V G ( K n × H ) E η Figure 4: An example of E η . In this example, η = (1 , , , ∈ { , } V ( H ) . Four grey areas represent V i,η ( i ) for i = 1 , , , { , } V ( H ) with { , , . . . , | V ( H ) | − } ⊆ F q in the following way. Regard V ( H ) = { , . . . , k − } ⊆ F q for k = | V ( H ) | and consider the mapping { , } V ( H ) (cid:51) η (cid:55)→ (cid:80) i ∈ V ( H ) i η ( i ) ∈{ , . . . , | V ( H ) | − } ⊆ F q . This mapping is bijection and thus we can regard η as an element of F q .26or η ∈ { , . . . , | V ( H ) | − } , let δ η : F q → F q be a degree-(2 | V ( H ) | −
1) polynomial such that δ η ( z ) = z = η, z (cid:54) = η and 0 ≤ z < | V ( H ) | , arbitrary otherwise . Then, define a function ˜ x : F q → F E ( K n/ × H ) q by ˜ x ( · ) := (cid:80) η ∈{ , } V ( H ) δ η ( · ) x [ E η ]. Note that ˜ x satisfies (1) ˜ x ( η ) = x [ E η ] holds for all η ∈ { , } V ( H ) , and (2) for each e ∈ E ( K n/ × H ), thefunction F q (cid:51) z (cid:55)→ ˜ x ( z )[ e ] ∈ F q is a polynomial of degree 2 | V ( H ) | −
1. In condition (1), we identified E η with E ( K n/ × H ). For each η , the function δ η can be constructed by O (2 | V ( H ) | | V ( H ) | ) fieldoperations using the fast univariate polynomial interpolation [Hor72]. Since our computationalmodel is O (log n )-Word RAM, we can perform any field operation on F q in constant time. Thus,the construction of ˜ x can be done in time O (2 | V ( H ) | | V ( H ) || E ( H ) | n ). Using ˜ x ( · ), we can rewritethe recursion formula (4) as EMBCOL n,H,q ( x ) = (cid:88) η ∈{ ,..., | V ( H ) | − } EMBCOL n/ ,H,q (˜ x ( η )) . (5)Note that EMBCOL n,H,q (˜ x ( · )) is a univariate polynomial of degree | E ( H ) | (2 | V ( H ) | − IP Now we present IP that verifies the statement “ EMBCOL n,H,q ( x ) = C ”. Suppose that n = 2 t . IP consists of t + 1 rounds. The verifier is given a vector x ∈ F E ( K n × H ) q (in our case, x is the edgeindicator of the input graph G ). At each round, the verifier updates the vector x and the constant C . In r -th round, the protocol proceeds as follows. Verifier.
When r = t + 1, check EMBCOL n,H,q ( x ) = C and halt. Prover.
Send a polynomial G ( · ) of degree at most | E ( H ) | (2 | V ( H ) | −
1) over F q to the verifier(here, G ( · ) is expected to be the univariate polynomial EMBCOL n/ ,H,q (˜ x ( · )), where ˜ x is thepolynomial vector constructed from x ). Verifier.
Check C = (cid:80) z ∈{ ,..., | V ( H ) | − } G ( z ). If not, reject. Otherwise, construct the polynomialvector ˜ x ( · ) using x . Sample i ∼ Unif( F q ) and update x ← ˜ x ( i ), C ← G ( i ) and n ← n/ EMBCOL n/ ,H,q (˜ x ( · )). Note that we canconstruct EMBCOL n/ ,H,q (˜ x ( · )) by evaluating EMBCOL n/ ,H,q (˜ x ( · )) at | E ( H ) | (2 | V ( H ) | − Emb ( H )col via (3). Thus, one can modify IP above such that theverifier asks the prover to solve Emb ( H )col for (log n ) O ( | E ( H ) | ) instances. In what follows, we analyzethis modified protocol. Let n be the size of the original input. In the beginning of r -th round, the size of x is | E ( K n × H ) | / r − = | E ( H ) | · t − r +1 . Thus, in the ( t + 1)-th round, the verifier runs in constant time. Anyother task of the verifier can be done in time n (log n ) O ( | E ( H ) | ) (the bottleneck is the simulation ofthe reduction of constructing EMBCOL n/ ,H,q (˜ x ( · )) to Emb ( H )col ).27 .2.2 Completeness and Soundness The perfect completeness of IP is easy: If the statement is true, an honest prover convinces theverifier with probability 1 by sending the polynomial EMBCOL n/ ,H,q (˜ x ( · )) at each round (recallthat the polynomial EMBCOL n,H,q (˜ x ( · )) satisfies the recurence formula (5)).Now we show the soundness. Proposition 6.1.
Let n = 2 t . If the statement “ EMBCOL n,H,q ( x ) = C ” is false, then for anyprovers, the verifier rejects with probability (cid:16) − Dq (cid:17) t − .Proof. We show Proposition 6.1 by induction on the number of rounds. Suppose that the statement“
EMBCOL n,H,q ( x ) = C ” is false. In the last t -th round, the verifier immediately reject (withprobability 1).Let D := | E ( H ) | (2 | V ( H ) | −
1) be the degree of
EMBCOL n/ ,H,q (˜ x ( · )). Fix 1 ≤ r < t and supposethat the verifier rejects with probability (1 − D/q ) t − r during ( r + 1)-th to t -th rounds. Consider r -th round with the assumption that the the statement “ EMBCOL n,H,q ( x ) = C ” is false. Then, anyprovers cheat with sending a polynomial G ( · ) that is not equal to EMBCOL n/ ,H,q (˜ x ( · )). It holdsthat Pr i ∼ Unif( F q ) (cid:2) G ( i ) (cid:54) = EMBCOL n/ ,H,q (˜ x ( i )) (cid:3) ≥ − Dq since both G and EMBCOL n/ ,H,q (˜ x ( · )) are degree at most D . Therefore, with probability 1 − D/q , the rounds proceeds to the next ( t + 1)-th round with a false statement. By the inductionassumption, the verifier rejects with probability at least (1 − D/q ) · (1 − D/q ) t − r = (1 − D/q ) t − r ,which completes the proof of Proposition 6.1. In IP , the prover is required to send a polynomial G that is expected to be EMBCOL n/ ,H,q (˜ x ( · )).As mentioned above, this can be reduced to solving m = polylog( n ) instances of Emb ( H )col .Moreover, by Theorem 2.5, each of the m instances can be reduced to polylog( n ) instances of( Emb ( H )col , G ( H ) n, / ). In other words, an honest prover can construct the polynomial EMBCOL n/ ,H,q ( · )by solving m polylog( n ) = polylog( n ) instances of the distributional problem ( Emb ( H )col , G ( H ) n, / ).Suppose that the prover has oracle access to a randomized heuristic algorithm that solves( Emb ( H )col , G ( H ) n, / ) with success probability 1 − (log n ) − L . Then, by the union bound, the proba-bility that the oracle outputs at last one wrong answer is at most polylog( n ) · (log n ) − L ≤ (log n ) − L if L = L ( H, L ) is sufficiently large.This completes the proof of Theorem 2.8. EMB ( K a,b ) By combining Theorems 2.7 and 2.8 and Lemma 5.1, we obtain an interactive proof system for
Emb ( K a,b ) as follows. Corollary 6.2 (IP for
Emb ( K a,b ) ) . Let H be a fixed graph. There is an O (log n ) -round interactiveproof system IP for the statement “ Emb ( K a,b ) ( G ) = C ” such that, given an input ( G, n, C ) , • The verifier accepts with probability if the statement is true (perfect completeness), while itrejects for any prover with probability at least / otherwise (soundness). In each round, the verifier runs in time n (log n ) O ( ab ) and sends (log n ) O ( ab ) instances of Emb ( K a,b ) .Furthermore, for any contant L , there exists a constant L = L ( a, b, L ) such that, if the proversolves ( Emb ( K a,b ) , K a,b,n ) with success probbalitiy − (log n ) − L , then the verifier accepts withprobability − (log n ) − L .Proof. For a given graph G , the verifier applies Lemma 5.1 and reduces Emb ( K a,b ) to solving m = O (1) instances of Emb ( K a,b )col .Then, the verifier solves each of the m instances G (cid:48) , . . . , G (cid:48) m of Emb ( K a,b )col using the reduction ofTheorem 2.5 with the help of the prover. Let C (cid:48) , . . . , C (cid:48) m be the values obtained by the reduction.Now, the verifier suffices to check that, for each i ∈ [ m ], the answer of the i -th instance G (cid:48) i of Emb ( K a,b )col is C (cid:48) i (if all of these m values are the correct one, then the verifier could solve theoriginal instance G of Emb ( K a,b ) ).To this end, run IP of Theorem 2.8 with letting H = K a,b . Here, an honest prover suf-fices to solve ( Emb ( K a,b ) , K a,b,n ) since Emb ( K a,b )col reduces to solving polylog( n ) instances of( Emb ( K a,b ) , K a,b,n ) by combining the reductions of Lemma 5.1 and Theorem 2.7. In this section, we provide a sufficient condition for a direct product theorem to hold. We willpresent a direct product theorem for any distributional problem that admits a selector. The notionof selector that we use in this paper is defined below.
Definition 7.1 ((Oracle) Selector; [Hir15]) . A randomized oracle algorithm S is said to be a selector from Π to a distributional problem (Π , D ) with success probability − δ if1. given access to two oracles A , A one of which solves (Π , D ) with success probability − δ ,on input x , the algorithm S A ,A computes Π( x ) with high probability (say, probability ≥ ),and2. for any n ∈ N and any input x ∈ supp ( D n ) , each query q of S to the oracles A and A satisfies that q ∈ supp ( D n ) . In order to obtain a direct product theorem in the settings of fine-grained complexity, it will becrucial to consider a selector with polylog( n ) queries. In this subsection, we show the existence of a selector with polylog( n ) queries for Emb ( H )col . Wefirst recall the notion of instance checker, which is known to imply the existence of selector ([Hir15]). Definition 7.2 (Instance Checker; Blum and Kannan [BK95]) . For a problem Π , a randomizedoracle algorithm M is said to be an instance checker for Π if, for every instance x of Π and anyoracle A ,1. Pr M [ M A ( x ) = Π( x )] = 1 if A solves Π correctly on every input, and2. Pr M [ M A ( x ) (cid:54)∈ { Π( x ) , fail } ] = o (1) . n )queries for Emb ( H )col . Theorem 7.3.
There exists an instance checker
Checker for
Emb ( H )col such that, given a graph G ⊆ K n × H ,1. Checker runs in time (cid:101) O ( n ) ,2. for any oracle A , Checker A calls the oracle A at most (log n ) C H times, where C H is a constantthat depends only on H , and3. each query G (cid:48) of Checker satisfies G (cid:48) ⊆ K n × H .Proof. Recall IP of Theorem 2.8. For a given oracle A , Checker obtains C := A ( G ) and then runs IP using A as a prover to verify Emb ( H )col ( G ) = C . If the verifier accepts, then Checker outputs C and otherwise it outputs fail .Suppose the oracle A solves Emb ( H )col correctly. Then, Checker output the correct answer withprobability 1 by the perfect completeness of IP .Now we check the second condition of Definition 7.2. If A ( G ) is the correct answer, thenthe output of Checker is either A ( G ) or fail . Otherwise, IP proceeds with the false statement that Emb ( H )col ( G ) = A ( G ). It follows from the soundness of IP (c.f. Proposition 6.1) that Verifier rejectswith probability (1 − O ( q − )) t . Hence, Pr[ Checker ( G ) = fail ] ≥ − O ( t/q ) = 1 − o (1). Theorem 7.4 (Restatement of Theorem 2.9) . Let H be a fixed graph. There exists a selector S from Emb ( H )col to ( Emb ( H )col , G ( H ) n, / ) with success probability − / polylog( n ) such that1. S runs in time (cid:101) O ( n ) , and2. S makes at most polylog( n ) queries.Proof. We combine the instance checker C of Theorem 7.3 with a worst-case to average-case re-duction R (Theorem 2.5).Here is the algorithm of a selector S . Given a graph G and oracle access to A , A , for each b ∈ { , } , the selector S simulates the instance checker C ( G ), and answer any query q of theinstance checker by running the reduction R A b ( q ). If the checker outputs some answer other than fail , the selector S outputs the answer and halts.The correctness of S can be shown as follows. Let A b be an oracle that solves ( Emb ( H )col , G ( H ) n, / )with success probability 1 − / polylog( n ), where b ∈ { , } . By the correctness of the reduction R ,the algorithm R A b solves Emb ( H )col with high probability. Therefore, if the instance checker C issimulated with oracle access to R A b , by the property of an instance checker, C outputs the correctanswer with high probability. Moreover, C outputs a wrong answer with probability at most o (1);thus, the selector outputs the correct answer with high probability. Corollary 7.5.
There is an (cid:101) O ( n ) -time selector S from Emb ( K a,b ) to ( Emb ( K a,b ) , K a,b,n ) withsuccess probability − / polylog( n ) . Moreover, S makes at most polylog( n ) queries. roof. From Theorem 7.4, we obtain a selector S from Emb ( K a,b )col to ( Emb ( K a,b )col , G ( K a,b ) n, / ). Here,we let H = K a,b . Invoke Proposition 2.6 that reduces ( Emb ( K a,b )col , G ( K a,b ) n, / ) to ( Emb ( K a,b ) , K a,b,n )in 2 O ( a + b ) = O (1) time. Note that the reduction of Proposition 2.6 preserves the success probabilitywithin a constant factor. Thus, each oracle query of S can be replaced by the reduction andwe obtain a selector from Emb ( K a,b )col to ( Emb ( K a,b ) , K a,b,n ). By Lemma 5.1, Emb ( K a,b ) isefficiently reducible to Emb ( K a,b )col , from which the existence of a selector from Emb ( K a,b ) to( Emb ( K a,b ) , K a,b,n ) follows. Using the notion of selector, we provide a direct product theorem in the context of fine-grainedcomplexity. A direct product of a distributional problem is formally defined as follows.
Definition 7.6 (Direct Product) . Let k : N → N be any function, and (Π , D ) be any distributionalproblem. The k -wise direct product of (Π , D ) , denoted by (Π , D ) k , is defined as the distributionalproblem (Π k , D k ) such that1. ( D k ) n := D k ( n ) n for each n ∈ N , and2. Π k ( x , · · · , x k ( n ) ) := (Π( x ) , · · · , Π( x k ( n ) )) for any ( x , · · · , x k ( n ) ) ∈ supp ( D k ( n ) n ) . The following direct product theorem gives an almost uniform direct product, in the sense thatit requires O (log 1 /(cid:15) ) bits of non-uniform advice in order to identify which is a correct algorithm.We observe that the direct product theorem is quite efficient and useful even in the setting offine-grained complexity. Theorem 7.7 (Almost Uniform Direct Product; Impagliazzo, Jaiswal, Kabanets, and Wigder-son [IJKW10]) . Let k ∈ N , (cid:15), δ > be parameters that satisfy (cid:15) > exp ( − Ω( δk )) . There exists a ran-domized oracle algorithm M that, given access to an oracle C that solves (Π , D ) k with success prob-ability (cid:15) , with high probability, produces a list of deterministic oracle algorithms M , · · · , M m suchthat M Ci computes (Π , D ) with success probability δ for some i ∈ { , . . . , m } , where m = O (1 /(cid:15) ) .If an oracle C can be computed in T C ( n ) time, then the running time of M Ci is at most T C ( n ) · O ((log 1 /δ ) /(cid:15) ) for any i ; the running time of M is at most O ( T C ( n ) /(cid:15) ) .Proof Sketch. The algorithm M C operates as follows. Fix an instance size n ∈ N . Let U denote supp ( D n ). Repeat the following m = O (1 /(cid:15) ) times. Pick a random (ordered) subset B ⊂ U of size k , and pick a random ordered subset A ⊂ B of size k/
2. Evaluate C ( B ) and let v be the answersgiven by C ( B ) for the instances in A . Output an oracle algorithm M A,v defined below.The algorithm M CA,v is defined as follows. On input x ∈ U , check whether x = a i for some a i ∈ A ; if so, output v i . Otherwise, repeat the following O ((log 1 /δ ) /(cid:15) ) times. Sample a randomset B ⊃ A ∪ { x } of size k . (The randomness used here can be supplied by M C and thus M CA,v canbe made deterministic.) If the answers given by C ( B ) for the instances in A coincide with v , thenoutput the answer given by C ( B ) for the instance x and halt; otherwise, go to the next loop. Lemma 7.8.
Let (Π , D ) be a distributional problem. Suppose there exists a selector S from Π to (Π , D ) with success probability δ that calls an oracle at most polylog( n ) times. Let M , . . . , M m be alist of deterministic algorithms such that, (1) for some i ∈ { , . . . , m } , M i solves (Π , D ) with successprobability δ , (2) for all i ∈ { , . . . , m } , M i runs in time t M ( n ) , and (3) for all i, j ∈ { , . . . , m } , S M i ,M j runs in time t S ( n ) (here, t S ( n ) does not take the running times of M i and M j into account). hen, there exists a t ( n ) -time randomized algorithm that solves Π with high probability, where t ( n ) ≤ (cid:101) O ( m ( t M ( n ) + t S ( n )) log(1 /δ ) log m ) .Proof. Let x be an input. From the assumption, there exists a selector S such that Pr S [ S A ,A ( x ) =Π( x )] ≥ − / m for any oracles A , A and any input x . Here, at least one of A and A solves(Π , D ) with success probability δ . Note that the success probability can be assumed to be 1 − / m because one can repeat the computation of ( S, P ) O (log m ) times.We present a randomized algorithm B that solves Π. For each i, j ∈ { , . . . , m } , B runs S M i ,M j ( x ) and let c ij be its output. If there exists i ∈ { , . . . , m } such that c ij = c for all j ∈ { , . . . , m } , B outputs c . Repeat this procedure for O (log 1 /δ ) times. If B outputs nothingduring the iteration, B outputs anything. Since the overall running time of S M i ,M j is at most t S ( n )+ t M ( n ) polylog( n ) for every i, j , the algorithm B runs in time (cid:101) O ( m ( t M ( n ) + t S ( n )) log(1 /δ ) log m ).We claim the correctness of the algorithm B . From the assumption, there exists i ∈ { , . . . , m } such that M i solves (Π , D ) with success probability δ . Since we repeat the procedure O (log 1 /δ )times, this event holds with high probability. Under this event, by the property of the selector S ,we have Pr S [ S M i ,M j ( x ) = Π( x )] ≥ − / m for all j . By the union bound, with probability atleast 15 / c i,j = Π( x ) for every j . Similarly, we also have c j,i = Π( x ) for every j with probabilityat least 15 /
16. These two properties guarantee that the output of B is equal to Π( x ). Overall,with probability at least 1 − /
16, the algorithm B outputs Π( x ).We now present a completely uniform direct product theorem for any problem that admits apolylog( n )-query selector. Theorem 7.9 (Direct Product Theorem for Any Problem with Selector) . Let k ∈ N , (cid:15), δ > beparameters that satisfy (cid:15) > exp ( − Ω( δk )) . Let (Π , D ) be a distributional problem. Suppose thereexists a t S ( n ) -time selector S from Π to (Π , D ) with success probability δ that calls an oracle atmost polylog( n ) times.Suppose that there exists a t ( n ) -time heuristic algorithm solving (Π , D ) k with success prob-ability (cid:15) . Then, there exists a t (cid:48) ( n ) -time algorithm that solves Π with high probability. Here t (cid:48) ( n ) ≤ (cid:101) O (( t S ( n ) + t ( n )) log(1 /δ ) log(1 /(cid:15) ) /(cid:15) ) .Proof. Let A be a t ( n )-time heuristic algorithm solving (Π , D ) k with success probability (cid:15) . Byusing the algorithm M of Theorem 7.7, M A produces a list of oracle algorithms M , · · · , M m such that M Ai computes (Π , D ) with success probability 1 − δ for some i ∈ { , . . . , m } , where m = O (1 /(cid:15) ). Then, we apply Lemma 7.8 using M A , . . . , M Am as the list of algorithms. Note that,from Theorem 7.7, each M Ai runs in time t ( n ). Thus, the algorithm solving Π of Lemma 7.8 runsin time (cid:101) O (( t ( n ) + t S ( n )) log(1 /δ ) log(1 /(cid:15) ) /(cid:15) ).Combining the existence of a selector (Corollary 7.5) and the direct product theorem (Theo-rem 7.9), we obtain the main lower bound result of this paper (Theorem 1.2). Proof of Theorem 1.3.
We prove the contrapositive. Assume that there exists a t ( n )-time heuristicalgorithm that solves ( Emb ( K a,b ) , K a,b,n ) k with success probability (cid:15) := n − α/ , where t ( n ) = n a − α .By Corollary 7.5, there exists a (cid:101) O ( n )-time selector using polylog( n ) queries from Emb ( K a,b ) to ( Emb ( K a,b ) , K a,b,n ) with success probability δ := 1 − (log n ) − C H , where C H > H . We choose k = O ((log 1 /(cid:15) ) /δ ) ≤ O ( α log n ) large enough so that the assump-tion of Theorem 7.9 is satisfied. By Theorem 7.9, we obtain a t (cid:48) ( n )-time algorithm that solves Emb ( K a,b ) , where t (cid:48) ( n ) = (cid:101) O (( n + t ( n )) · n α/ ) ≤ (cid:101) O ( n a − α/ ). This contradicts Theorem 2.1.32 Fine-Grained XOR Lemma
In this section, we show a XOR lemma in the context of fine-grained complexity. We focus on the
XOR problem Π ⊕ k defined as follows. Definition 8.1.
Let Π be a problem such that Π( x ) ∈ { , } for any input x . For a parameter k ∈ N , let Π ⊕ k be the problem of computing (cid:80) ki =1 Π( x i ) (mod 2) on input ( x , . . . , x k ) . Throughout this section, we consider decision problems unless otherwise noted. For a distribu-tional problem (Π , D ), let D k be the direct product of D (see Definition 7.6). Suppose that thereis a selector from Π to (Π , D ) that makes at most polylog( n ) queries. The aim of this section isto derive the average-case hardness of the distributional problem (Π ⊕ k , D k ) from the worst-casehardness assumption of Π (see Theorem 8.3). To this end, we combine Direct Product Theorem(Theorem 7.9) and the well-known list-decoding technique for the Hadamard code due to Goldreichand Levin [GL89]. Let us restate the Goldreich-Levin theorem as follows. Theorem 8.2 (Goldreich-Levin Theorem [GL89]) . Let (Π , D ) be a distributional problem and let k = k ( n ) ∈ N , (cid:15) = (cid:15) ( n ) > be parameters. Then, there exists an algorithm M that, given accessto an oracle A solving (Π ⊕ k , D k ) with success probability / (cid:15) , produces with high probability alist of deterministic oracle algorithms M , . . . , M m such that, for some t ∈ { , . . . , m } , the oraclealgorithm M At solves (Π , D ) k with success probability / . Here, m = O ( k/(cid:15) ) .If an oracle A can be computed in time T A ( n ) , then each M Ai runs in time O (cid:0) T A ( n ) k . /(cid:15) (cid:1) forany i , and M runs in time O (cid:0) m · T A ( n ) k . /(cid:15) (cid:1) = O (cid:0) T A ( n ) k . /(cid:15) (cid:1) .Proof. Let (Π , D ) be the distributional problem and k = k ( n ) , (cid:15) = (cid:15) ( n ) be the parameters mentionedin Theorem 8.2. Consider the following problem Π (cid:48) : Given 2 k instances x , . . . , x k of Π and r ∈ { , } k , compute (cid:80) ki =1 r i · Π( x i ) mod 2. Let (Π (cid:48) , D (cid:48) ) be a distributional problem, where, in D (cid:48) ,the input is sampled as ( x , . . . , x k ) ∼ D k and r ∼ Unif( { , } k ). Note that, if r ∼ Unif( { , } k ),with probability at least (cid:0) kk (cid:1) / k ≥ / (2 √ k ), the vector r ∈ { , } k has exactly k ones. Here, weused the well-known inequality (cid:0) kk (cid:1) ≥ (cid:0) − k (cid:1) k √ πk . Conditioned on this event, the distributionalproblem (Π (cid:48) , D (cid:48) ) is equivalent to (Π ⊕ k , D k ). Let A (cid:48) be the algorithm that takes 2 k instances x , . . . , x k and r ∈ { , } k as input and outputs A ( x i , . . . , x i k ) if r contains exactly k ones in theposition of i < · · · < i k ; otherwise outputs a random bit. This algorithm A (cid:48) runs in time O ( t ( n ))and solves (Π (cid:48) , D (cid:48) ) with success probability at least 1 / (cid:15)/ (2 √ k ). In other words,Pr A (cid:48) ,x ,...,x k ,r ∼ Unif( { , } k ) (cid:34) A (cid:48) ( x , . . . , x k , r ) = k (cid:88) i =1 r i Π( x i ) mod 2 (cid:35) ≥
12 + (cid:15) √ k . Now we present the algorithm M mentioned in Theorem 8.3. We say that an input ( x , . . . , x k )is good if Pr A (cid:48) ,r ∼ Unif( { , } k ) (cid:34) A (cid:48) ( x , . . . , x k , r ) = k (cid:88) i =1 r i Π( x i ) mod 2 (cid:35) ≥
12 + (cid:15) √ k . We claim that at least (cid:15)/ (4 √ k ) fraction of ( x , . . . , x k ) are good. To see this, let E be the event that A (cid:48) success (i.e., A (cid:48) ( x , . . . , x k , r ) = (cid:80) ki =1 r i Π( x i ) (mod 2)) and let F be the event that ( x , . . . , x k )33s good. Assume Pr[ F ] < (cid:15)/ (4 √ k ). Then, from the property of A (cid:48) and the assumption, we have12 + (cid:15) √ k ≤ Pr[ E ] ≤ Pr[
E|F ] Pr[ F ] + Pr[ E| not F ] Pr[not F ] < (cid:15) √ k + (cid:18)
12 + (cid:15) √ k (cid:19) = 12 + (cid:15) √ k . Thus we have Pr[ F ] = Pr[( x , . . . , x k ) is good] ≥ (cid:15)/ (4 √ k ).Let m = 24 k/(cid:15) and (cid:96) be the minimum integer satisfying m ≤ (cid:96) . The algorithm M producesa list M , . . . , M (cid:96) such that, for some i ∈ { , . . . , (cid:96) } , M A (cid:48) i solves (Π , D ) k for good inputs. Let s (1) , . . . , s ( (cid:96) ) ∼ Unif( { , } k ) be (cid:96) i.i.d. random vectors. Construct m distinct nonempty subsets T , . . . , T m ⊆ [ (cid:96) ] in a canonical way and let r ( i ) := (cid:80) j ∈ T i s ( i ) j . Note that, for every i (cid:54) = i (cid:48) , r ( i ) and r ( i (cid:48) ) are pairwise independent random vectors and each r ( i ) is drawn from Unif( { , } k ).Now we present the list of oracle algorithms M , . . . , M (cid:96) . For each t ∈ { , . . . , (cid:96) } , the algorithm M A (cid:48) t works as follows. Write t = (cid:80) (cid:96)j =1 j − w j as a binary extension. In other words, ( w , . . . , w (cid:96) )can be seen as an (cid:96) -bits of advice. The bit w j tells us the value of (cid:104) Π( x ) , s ( j ) (cid:105) := (cid:80) ki =1 Π( x i ) s ( j ) i (mod 2). Note that, for some t , this equality holds for all j = 1 , . . . , (cid:96) .Suppose that the input ( x , . . . , x k ) is good. Given ( w , . . . , w (cid:96) ), for every i = 1 , . . . , m , M A (cid:48) t does the following: First, M A (cid:48) t computes W ( i ) := (cid:80) j ∈ T i w j . Note that, for some t , we have W ( i ) = (cid:80) j ∈ T i (cid:104) Π( x ) , s ( j ) (cid:105) = (cid:104) Π( x ) , r ( i ) (cid:105) . Then, for every index l ∈ { , . . . , k } , M A (cid:48) t calls the oracleand obtain A (cid:48) ( x , . . . , x k , r ( i )1 , . . . , r ( i ) l − , r ( i ) l , r ( i ) l +1 , . . . , r ( i )2 k ), where z := 1 − z for z ∈ { , } . Theoutput O ( i ) satisfies W ( i ) + O ( i ) = Π( x l ) (mod 2) if A (cid:48) success. This happens with probability1 / (cid:15)/ (4 √ k ) since ( x , . . . , x k ) is good. We repeat this for Q = 96 k . /(cid:15) times and then we cancompute Π( x l ) by taking the majority among the Q trials with successes probability at least 1 − k for each l = 1 , . . . , k . To see this, let Z i be a binary indicator random variable such that Z i = 1 ifand only if W ( i ) + O ( i ) = Π( x l ). Let Z = Z + · · · + Z Q . It suffices to show Pr[ Z > Q/ ≥ − k .Note that E [ Z ] ≥ Q + (cid:15)Q √ k and Var [ Z ] = (cid:80) Qi =1 Var [ Z i ] ≤ Q since the random variables Z i arepairwise independent. From the Chebyshev inequality, we obtainPr (cid:20) Z ≤ Q (cid:21) ≤ Pr (cid:20) | Z − E [ Z ] | ≥ (cid:15)Q √ k (cid:21) ≤ Pr (cid:20) | Z − E [ Z ] | ≥ (cid:15) √ Q √ k (cid:112) Var [ Z ] (cid:21) ≤ √ k(cid:15) Q ≤ k . Here, recall that the Chebyshev inequality assertsPr (cid:104) | Z − E [ Z ] | ≥ ξ (cid:112) Var [ Z ] (cid:105) ≤ ξ for any ξ >
0. Then, from the union bound over 2 k indices, M A (cid:48) t (for the appropriate t ) computes(Π( x ) , . . . , Π( x k )) with probability at least 2 / M A (cid:48) i is deterministic without loss of generality since the coin flips can be given by M . The success probability of M A (cid:48) i is at least (2 / · ( (cid:15)/ (4 √ k )) ≥ (cid:15)/ (6 √ k ) since input ( x , . . . , x k )is good with probability at least (cid:15)/ (4 √ k ). The running time of M A (cid:48) i is O ( Qk ) = O ( k . /(cid:15) ) for all34 ∈ { , . . . , m } . Thus, if A (cid:48) is a T A ( n )-time algorithm, then we can construct M i as a deterministic O ( T A ( n ) k . /(cid:15) )-time algorithm. The total running time of M is at most m · O ( T A ( n ) k . /(cid:15) ) since M constructs m = O ( k/(cid:15) ) algorithms each of them runs in time O ( T A ( n ) k . /(cid:15) ).Now we prove the main result of this section. Theorem 8.3 (XOR Lemma for Any Problem with Selector) . Let k ∈ N , (cid:15), δ > be parameterssatisfying (cid:15) > exp( − Ω( δk )) . Let (Π , D ) be a distributional decision problem. Suppose there existsa t S ( n ) -time selector S from Π to (Π , D ) with success probability δ that calls an oracle at most polylog( n ) times.Suppose that there exists a t ( n ) -time heuristic algorithm solving (Π ⊕ k , D k ) with success prob-ability + (cid:15) . Then, there exists a t (cid:48) ( n ) -time randomized algorithm that solves Π with probability / . Here t (cid:48) ( n ) ≤ (cid:101) O (cid:0) ( t S ( n ) + t ( n )) · log(1 /δ )( k/(cid:15) ) (cid:1) .Proof. From the assumption, we have a t ( n )-time heuristic algorithm A solving (Π ⊕ k , D k ) withsuccess probability (cid:15) . Then, from Theorem 8.2 with using A as the oracle, we obtain a list ofdeterministic oracle algorithms M , . . . , M m such that M Ci solves (Π , D ) k with success probabilityΩ (cid:0) (cid:15)/ √ k (cid:1) for some i ∈ { , . . . , m } , where m = O ( k/(cid:15) ). Each of M i runs in time O ( t ( n ) k . /(cid:15) ) ifwe take the running time of C into account. This list can be constructed in time O ( t ( n ) k . /(cid:15) ).Let δ > i ∈ { , . . . , m } , apply The-orem 7.9 using M i as the oracle C mentioned in Theorem 7.9. This yields a list M i, , . . . , M i,m (cid:48) of deterministic algorithms for each i ∈ { , . . . , m } , where m (cid:48) = O (1 /(cid:15) ). Moreover, if M i ∗ solves(Π , D ) k , then M i ∗ ,j ∗ solves Π with success probability δ for some j ∗ ∈ { , . . . , m (cid:48) } . For every i, j , M i,j runs in time O ( t ( n ) k . /(cid:15) · (log 1 /δ ) /(cid:15) ) ≤ O ( t ( n ) k . log(1 /δ ) /(cid:15) ).Now we have a list ( M i,j ) of mm (cid:48) = O ( k/(cid:15) ) deterministic algorithms. From Lemma 7.8,there exists an algorithm B that solves Π with high probability. The overall running time of B is at most (cid:101) O (cid:0) ( mm (cid:48) ) ( t S ( n ) + t ( n ) k . log(1 /δ ) /(cid:15) ) log( √ k/(cid:15) ) log( mm (cid:48) ) (cid:1) ≤ (cid:101) O (cid:0) ( t S ( n ) + t ( n )) · log(1 /δ )( k/(cid:15) ) (cid:1) . ⊕ EMB ( H )col Let H be a fixed graph. Consider the problem ⊕ Emb ( H )col of computing the parity of Emb ( G )col ( H )for a given graph G . For a parameter k , let (cid:85) k G ( H ) n, / be the distribution of random graphsthat is a direct sum of k i.i.d. graphs drawn from G ( H ) n, / . That is, let G , . . . , G k ∼ G ( H ) n, / bei.i.d. random graphs. Suppose that G ( V i ) ∩ G ( V j ) = ∅ for any i (cid:54) = j . Then, the graph G definedby V ( G ) = (cid:83) ki =1 V ( G i ) and E ( G ) = (cid:83) ki =1 E ( G i ) forms the distribution (cid:85) k G ( H ) n, / . Let Emb ( H )col bethe decision problem in which we are asked to decide whether Emb ( H )col ( G ) > G . This subsection is devoted to the following result. Theorem 8.4.
Suppose that there exists a t ( n ) -time heuristic algorithm solving the distributionalproblem ( ⊕ Emb ( H )col , (cid:85) k G ( H ) n, / ) with success probability + (cid:15) far any k = O (log (cid:15) − ) . Then, thereexists a t ( n ) · (log n/(cid:15) ) O (1) -time randomized algorithm solving Emb ( H )col with probability / . The proof of Theorem 8.4 consists of the following three steps. First, we present a randomizedreduction of
Emb ( H )col to ⊕ Emb ( H )col in the worst-case sense. Then, we check that the parity problem ⊕ Emb ( H )col admits a (cid:101) O ( n )-time selector with polylog( n ) queries. Finally, we apply Theorem 8.3 toboost the error tolerance. The second and third steps imply Theorem 2.10. More specifically, weobtain the following. 35 heorem 8.5 (Refinement of Theorem 2.10) . Let H be an arbitrary graph. Suppose that thereexists a T ( n ) -time randomized heuristic algorithm that solves ( ⊕ Emb ( H )col , (cid:85) k G ( H ) n, / ) with successprobability greater than + (cid:15) for any k = O (log (cid:15) − ) . Then, there exists a T ( n )(log n/(cid:15) ) O (1) -timerandomized algorithm that solves ⊕ Emb ( H )col with probability at least / for any input.Remark . Theorem 2.10 immediately follows from Theorem 8.5 (substitute (cid:15) = n − c to Theo-rem 8.5). Parity vs. Detection.Lemma 8.7.
Suppose that there exists a t ( n ) -time randomized algorithm solving ⊕ Emb ( H )col for anyinput with probability at least / . Then, there exists a t (cid:48) ( n ) -time randomized algorithm that solves Emb ( H )col with probability at least / . Here, t (cid:48) ( n ) = O (2 | E ( H ) | t ( n )) .Proof. The proof is essentially given in Appendix A of [BABB19]. For completeness, we presentthe proof. Consider the polynomial P G : F E ( G )2 → F defined as P G ( x ) := (cid:88) F ⊆ E ( G ): F is isomorphic to H (cid:89) e ∈ F x e . Then, G does not contain H if and only if P G ( · ) ≡
0. The degree of P G is | E ( H ) | . Moreover, if P G ( · ) (cid:54)≡
0, then P G ( z ) = 1 for at least 2 −| E ( H ) | fraction of z ∈ F | E ( G ) | (see, e.g., Lemma 2.6 of[NS94]).Now we present an algorithm that solves Emb ( H )col using an oracle that solves ⊕ Emb ( H )col . Let m = 100 · | E ( H ) | and sample m i.i.d. random vectors z , . . . , z m ∼ Unif( F E ( G )2 ). Then, compute P G ( z ) , . . . , P G ( z m ). If P G ( z i ) = 1 for some i , output YES. Otherwise, output NO. Note that onecan compute P G ( · ) by solving ⊕ Emb ( H )col since P G ( · ) is a polynomial over F .If G does not contain H , the algorithm outputs NO with probability 1. If G contains H , theprobability that the algorithm outputs NO is at most (1 − −| E ( H ) | ) m ≤ e − . Selector for ⊕ EMB ( H )col .Theorem 8.8. There exists a selector S from ⊕ Emb ( H )col to ( ⊕ Emb ( H )col , G ( H ) n, / ) with success proba-bility − / polylog( n ) such that (1) S runs in time (cid:101) O ( n ) , and (2) The number of oracle accessesis at most polylog( n ) .Proof. The proof is basically the same as that of Theorem 7.4. To construct a worst-case-to-average-case reduction R (cid:48) for ⊕ Emb ( H )col , we encode ⊕ Emb ( H )col to the low-degree polynomial EMBCOL n,H, F t .Note that, since F t has characteristic 2 (i.e., a + a = 0 for any a ∈ F t ), computing thepolynomial EMBCOL n,H, F t ( x ) for x ∈ { , } E ( H ) × K n is equivalent to solving ⊕ Emb ( H )col by re-garding the input x as the edge indicator of a graph. Using Corollary 4.2, we reduce comput-ing EMBCOL n,H, F t to solving the distributional problem ( EMBCOL n,H, F t , U ( H ) n ( F t )). Moreover,we can reduce ( EMBCOL n,H, F t , U ( H ) n ( F t )) to ( EMBCOL n,H, F t , U ( H ) n ( F )) with query complexity(log n ) O ( | E ( H ) | ) using the technique of [BABB19]. This yields a worst-case-to-average-case reductionfor ⊕ Emb ( H )col (c.f., Theorem 2.5). 36imilarly, a slight modification of the interactive proof system IP of Theorem 2.8 works for ⊕ Emb ( H )col . To be more specifically, let us consider an interactive proof system IP (cid:48) for the state-ment “ ⊕ Emb ( H )col ( G ) = b ”. The protocol IP (cid:48) is the same as IP except for using F t instead of F q .Note that the equation (5) holds even for EMBCOL n,H, F t . Moreover, computing the polynomial EMBCOL n,H, F t can be reduced to computing EMBCOL n,H, F using the aforementioned techniqueof [BABB19].Using the interactive proof system IP (cid:48) for ⊕ Emb ( H )col , we can construct an (cid:101) O ( n )-time polylog( n )-query instance checker C (cid:48) for ⊕ Emb ( H )col (see Theorem 7.3). Combining the instance checker C (cid:48) andthe worst-case-to-average-case reduction R (cid:48) , we can construct the desired selector (see the proof ofTheorem 7.4). XOR lemma for ⊕ EMB ( H )col (proof of Theorem 8.5). Assume that there exists a t ( n )-timeheuristic algorithm A that solves ( ⊕ Emb ( H )col , (cid:85) k G ( H ) n, / ) with success probability (cid:15) . Note thatthe distributional problem ( (cid:76) ( ⊕ Emb ( H )col ) k , ( G ( H ) n, / ) k ) is equivalent to the distributional problem( ⊕ Emb ( H )col , (cid:85) k G ( H ) n, / ). Hence, the algorithm A also solves ( (cid:76) ( ⊕ Emb ( H )col ) k , ( G ( H ) n, / ) k ). From The-orem 8.8, there exists an (cid:101) O ( n )-time selector using polylog( n ) oracle accesses from ⊕ Emb ( H )col to( ⊕ Emb ( H )col , G ( H ) n, / ) with success probability δ := 1 − (log n ) − C for some constant C > H . Let k = k ( n ) be the parameter such that the assumptions of Theorem 8.3is satisfied. Since δ = 1 − (log n ) − C , we can set k = O (log (cid:15) − ). Then, by Theorem 8.3, wehave an t (cid:48) ( n )-time ranndomized algorithm that solves ⊕ Emb ( H )col with high probability, where t (cid:48) ( n ) = (cid:101) O (( n + t ( n )) · ( k/(cid:15) ) = t ( n ) · (log n/(cid:15) ) O (1) (here we assume t ( n ) ≥ n ). Proof of Theorem 8.4.
We combine Lemma 8.7 and Theorem 8.5. Suppose that there existsa t ( n )-time heuristic algorithm solving ( ⊕ Emb ( H )col , (cid:85) k G ( H ) n, / ) with success probability (cid:15) . FromTheorem 8.5, there exists a t ( n ) · (log n/(cid:15) ) O (1) -time randomized algorithm for ⊕ Emb ( H )col . Then,from Lemma 8.7, we obtain a 2 | E ( H ) | · t ( n ) · (log n/(cid:15) ) O (1) -time randomized algorithm for Emb ( H )col . ⊕ K a - Subgraph
Recall that ⊕ K a - Subgraph is the problem of computing the parity of the number of K a subgraphscontained in a given graph. This subsection is devoted to prove Theorem 1.4. Recall that G n, / isthe distribution of the Erd˝os-R´enyi graph G ( n, / (cid:85) k G n, / is the distribution of the disjointunion of k random graphs G , . . . , G k each of which is independently drawn from G n, / . Theorem 8.9 (Refinement of Theorem 1.4) . Suppose that there exists a T ( n ) -time randomizedheuristic algorithm that solves ( ⊕ K a - Subgraph , (cid:85) k G n, / ) with success probability + ε for any k = O (log ε − ) . Then, there exists a T ( n )(log n/ε ) O (1) -time randomized algorithm that solves ⊕ K a - Subgraph for any input with probability / .Proof of Theorem 1.4. Theorem 8.9 directly implies Theorem 1.4 (let ε = n − (cid:15) ).The core of the proof of Theorem 8.9 is the existence of the following efficient selector. Lemma 8.10.
There exists an (cid:101) O ( n ) -time selector S from ⊕ K a - Subgraph to the distributionalproblem ( ⊕ K a - Subgraph , G n, / ) with success probability − / polylog( n ) . Moreover, the numberof oracle accesses of S is at most polylog( n ) . roof. The proof of Lemma 8.10 is similar to that of Corollary 7.5. From Theorem 8.8, we have ob-tain a selector from ⊕ Emb ( K a )col to ( ⊕ Emb ( K a )col , G ( K a ) n, / ). Then, we use the reduction by Boix-Adser´a,Brennan, and Bresler [BABB19]. They reduced ( ⊕ Emb ( K a )col , G ( K a ) n, / ) to ( ⊕ K a - Subgraph , G n, / )with preserving the success probability up to a constant factor (Lemma 3.10 of [BABB19]). Us-ing their reduction, each query of the selector S can be replaced by the reduction. This yields aselector from ⊕ Emb ( K a )col to ( ⊕ K a - Subgraph , G n, / ). We then use the reduction of Lemma 3.3of Boix-Adser´a, Brennan, and Bresler [BABB19]. They reduced ⊕ K a,b - Subgraph to ⊕ Emb ( K a )col .Specifically, if ⊕ Emb ( K a )col can be solved in time t ( n ), then there exists a t ( n )+ O ( n )-time algorithmfor ⊕ K a - Subgraph . Proof of Theorem 8.9.
Suppose that there exists a T ( n )-time randomized heuristic algorithm thatsolves ( ⊕ K a - Subgraph , (cid:85) k G n, / ) with success probability + (cid:15) for any k = O (log (cid:15) − ). Note that( ⊕ K a - Subgraph , (cid:85) k G n, / ) is equivalent to (( ⊕ K a,b - Subgraph ) ⊕ k , ( G n, / ) k ). From Theorem 8.3and Lemma 8.10, we obtain an t (cid:48) ( n )-time randomized algorithm that solves ⊕ K a,b - Subgraph with probability 2 /
3, where t (cid:48) ( n ) = ( n + t ( n )(log n/(cid:15) ) O (1) = t ( n ) · (log n/(cid:15) ) O (1) (here, we assume t ( n ) = Ω( n ) and let δ = 1 − (log n ) − C ) and k = O (log (cid:15) − )). Acknowledgement
We thank Marhsall Ball for helpful discussion on Proof of Work. Shuichi Hirahara is supported byACT-I, JST. Nobutaka Shimizu is supported by JSPS KAKENHI Grant Number JP19J12876.
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