Clique Is Hard on Average for Regular Resolution
Albert Atserias, Ilario Bonacina, Susanna F. de Rezende, Massimo Lauria, Jakob Nordström, Alexander Razborov
aa r X i v : . [ c s . CC ] D ec Clique Is Hard on Average for Regular Resolution
Albert Atserias , Ilario Bonacina , Susanna F. de Rezende , Massimo Lauria ,Jakob Nordstr ¨om and Alexander Razborov Universitat Polit`ecnica de Catalunya Institute of Mathematics of the Czech Academy of Sciences Sapienza - Universit`a di Roma University of Copenhagen and Lund University University of Chicago and Steklov Mathematical InstituteDecember 18, 2020
Abstract
We prove that for k ≪ √ n regular resolution requires length n Ω( k ) to establish that anErd˝os–R´enyi graph with appropriately chosen edge density does not contain a k -clique. Thislower bound is optimal up to the multiplicative constant in the exponent, and also implies uncon-ditional n Ω( k ) lower bounds on running time for several state-of-the-art algorithms for findingmaximum cliques in graphs. Deciding whether a graph has a k -clique is one of the most basic computational problems on graphs,and has been extensively studied in computational complexity theory ever since it appeared inKarp’s list of NP -complete problems [Kar72]. Not only is this problem widely believed tobe intractable to solve exactly (unless P = NP ), there does not even exist any polynomial-timealgorithm for approximating the maximum size of a clique to within a factor n − ǫ for any constant ǫ > , where n is the number of vertices in the graph [H˚as99, Zuc07]. Furthermore, the problemappears to be hard not only in the worst case but also on average in the Erd˝os-R´enyi random graphmodel—we know of no efficient algorithms for finding cliques of maximum size asymptoticallyalmost surely on random graphs with appropriate edge densities [Kar76, Ros10].In terms of upper bounds, the k -clique problem can clearly be solved in time roughly n k simplyby checking if any of the (cid:0) nk (cid:1) many sets of vertices of size k forms a clique. This takes polynomialtime if k is constant. This can be improved slightly to O( n ωk/ ) , where ω ≤ . is the matrix1ultiplication exponent, using algebraic techniques [NP85], although in practice such algebraicalgorithms are outperformed by combinatorial ones [Vas09].The motivating problem behind this work is to determine the exact time complexity of the cliqueproblem when k is given as a parameter. As noted above, all known algorithms require time n Ω( k ) .It appears quite likely that some dependence on k is needed in the exponent, since otherwise wehave the parameterized complexity collapse FPT = W [1] [DF95]. Even more can be said if weare willing to believe the Exponential Time Hypothesis (ETH) [IP01]—then the exponent has todepend linearly on k [CHKX04], so that the trivial upper bound is essentially tight.Obtaining such a lower bound unconditionally would, in particular, imply P = NP , and so cur-rently seems completely out of reach. But is it possible to prove n Ω( k ) lower bounds in restricted butnontrivial models of computation? For circuit complexity, this challenge has been met for circuitsthat are of bounded depth [Ros08] or are monotone [Ros14]. In this paper we focus on computa-tional models that are powerful enough to capture several algorithms that are used in practice.When analysing such algorithms, it is convenient to view the execution trace as a proof estab-lishing the maximum clique size for the input graph. In particular, if this graph does not have a k -clique, then the trace provides an efficiently verifiable proof of the statement that the graph is k -clique-free. If one can establish a lower bound on the length of such proofs, then this implies alower bound on the running time of the algorithm, and this lower bound holds even if the algorithmis a non-deterministic heuristic that somehow magically gets to make all the right choices. Thisbrings us to the topic of proof complexity [CR79], which can be viewed as the study of upper andlower bounds in restricted nondeterministic computational models.Using a standard reduction from k -clique to SAT, we can translate the problem of k -cliquesin graphs to that of satisfiability of formulas in conjunctive normal form (CNF). If an algorithmfor finding k -cliques is run on a graph G that is k -clique-free, then we can extract a proof of theunsatisfiability of the corresponding CNF formula—the k -clique formula on G —from the executiontrace of the algorithm. Is it possible to show any non-trivial lower bound on the length of suchproofs? Specifically, does the resolution proof system—the method of reasoning underlying state-of-the-art SAT solvers [BS97, MS99, MMZ + n Ω( k ) , or at least n ω k (1) (i.e. theexponent as a function of k is not bounded by a constant), to prove the absence of k -cliques in agraph? This question was asked in, e.g., [BGLR12] and remains open.The hardness of k -clique formulas for resolution is also a problem of intrinsic interest in proofcomplexity, since these formulas escape known methods of proving resolution lower bounds for arange of interesting values of k including k = O(1) . In particular, the interpolation technique [Kra97,Pud97], the random restriction method [BP96], and the size-width lower bound [BW01] all seemto fail.To make this more precise, we should mention that some previous works do use the size-widthmethod, but only for very large k . It was shown in [BIS07] that for n / ≪ k ≤ n/ resolution re-quires length exp (cid:0) n Ω(1) (cid:1) to certify that a dense enough Erd˝os-R´enyi random graph is k -clique-free.The constant hidden in the Ω(1) increases with the density of the graph and, in particular, for verydense graphs and k = n/ the length required is Ω( n ) . Also, for a specially tailored CNF encoding,where the i th member of the claimed k -clique is encoded in binary by log n variables, a lower boundof n Ω( k ) for k ≤ log n can be extracted from a careful reading of [LPRT17]. However, in the more2atural unary encodings, where indicator variables specify whether a vertex is in the clique, thesize-width method cannot yield more than a Ω( k /n ) lower bound since there are resolution proofsof width O( k ) . This bound becomes trivial when k ≤ √ n .In the restricted subsystem of tree-like resolution , optimal n Ω( k ) length lower bounds were es-tablished in [BGL13] for k -clique formulas on complete ( k − -partite as well as on average forErd˝os-R´enyi random graphs of appropriate edge density. There is no hope to get hard instances forgeneral resolution from complete ( k − -partite graphs, however—in the same paper it was shownthat all instances from the more general class of ( k − -colourable graphs are easy for resolution.A closer study of these resolution proofs reveals that they are regular , meaning that if the proof isviewed as a directed acyclic graph (DAG), then no variable is eliminated more than once on anysource-to-sink path.More generally, regular resolution is an interesting and non-trivial model to analyse for the k -clique problem since it captures the reasoning used in many state-of-the-art algorithms used inpractice (for a survey, see, e.g., [Pro12, McC17]). Nonetheless, it has remained consistent with state-of-the-art knowledge that for k ≤ n / regular resolution might be able to certify k -clique-freenessin polynomial length independent of the value of k . Our contributions
We prove optimal n Ω( k ) average-case lower bounds for regular resolutionproofs of unsatisfiability for k -clique formulas on Erd˝os-R´enyi random graphs. Theorem 1.1 (Informal) . For any integer k ≪ √ n , given an n -vertex graph G sampled at randomfrom the Erd˝os-R´enyi model with the appropriate edge density, regular resolution asymptoticallyalmost surely requires length n Ω( k ) to certify that G does not contain a k -clique. At a high level, the proof is based on a bottleneck counting argument in the style of [Hak85] witha slight twist that was introduced in [RWY02]. In its classical form, such a proof takes four steps.First, one defines a distribution of random source-to-sink paths on the DAG representation of theproof. Second, a subset of the vertices of the DAG is identified—the set of bottleneck nodes —suchthat any random path must necessarily pass through at least one such node. Third, for any fixedbottleneck node, one shows that it is very unlikely that a random path passes through this particularnode. Given this, a final union bound argument yields the conclusion that the DAG must have manybottleneck nodes, and so the resolution proof must be long.The twist in our argument is that, instead of single bottleneck nodes, we need to define bottleneckpairs of nodes. We then argue that any random path passes through at least one such pair but that fewrandom paths pass through any fixed pair; the latter part is based on Markov chain-type reasoningsimilar to [RWY02, Theorems 3.2, 3.5]. Furthermore, it crucially relies on the graph satisfying acertain combinatorial property, which captures the idea that the common neighbourhood of a smallset of vertices is well distributed across the graph. Identifying this combinatorial property is a keycontribution of our work. In a separate argument (that, surprisingly, turned out to be much moreelaborate than most arguments of this kind) we then establish that Erd˝os-R´enyi random graphs ofthe appropriate edge density satisfy this property asymptotically almost surely. Combining thesetwo facts yields our average-case lower bound.The idea of counting bottlenecks of more than one node comes from [RWY02] and was alsoused in [BBI16]. 3nother contribution of this paper is a relatively simple observation that not only is regularresolution powerful enough to distinguish graphs that contain k -cliques from ( k − -colourablegraphs [BGL13], but it can also distinguish them from graphs that have a homomorphism to anyfixed graph H with no k -cliques. Recent Developments
A preliminary version of this work appeared in the proceedings of theSTOC’18 conference [ABdR + n Ω( k ) average-case lowerbound for regular resolution were recently extended by Pang [Pan19] to work for a proof systembetween regular and general resolution. In the same paper, Pang also shows a Ω( k (1 − ǫ ) ) resolutionlower bound for k -clique formulas on Erd˝os-R´enyi random graphs, for k = n c , c < / and ǫ > .Regarding the proof complexity of k -clique formulas for tree-like resolution, the lower boundsfrom [BGL13] and [LPRT17] were simplified and unified in [Lau18]. The resolution lower boundin [LPRT17] for k -clique formulas on Erd˝os-R´enyi random graphs under the binary encoding wasrecently extended to an n Ω( k ) /d ( s ) lower bound for Res( s ) , where s = o ((log log n ) / ) and d ( s ) isa doubly exponential function [DGGM20]. Paper outline
The rest of this paper is organized as follows. Section 2 presents some preliminar-ies. We show that some nontrivial k -clique instances are easy for regular resolution in Section 3.Section 4 contains the formal statement of the lower bounds we prove for Erd˝os-R´enyi randomgraphs. In Section 5 we define a combinatorial property of graphs and show that clique formulason such graphs are hard for regular resolution, and the proof that Erd˝os-R´enyi random graphs sat-isfy this property asymptotically almost surely is in Section 6. Section 7 explains why our resultsimply lower bounds on the running time of state-of-the-art algorithms for k -clique. We concludein Section 8 with a discussion of open problems. We write G = ( V, E ) to denote a graph with vertices V and edges E , where G is always undirected,without loops and multiple edges. Given a vertex v ∈ V , we write N ( v ) to denote the set of neighbours of v . For a set of vertices R ⊆ V we write b N ( R ) = T v ∈ R N ( v ) to denote the setof common neighbours of R . For two sets of vertices R ⊆ V and W ⊆ V we write b N W ( R ) = b N ( R ) ∩ W to denote the set of common neighbours of R inside W . For a set U ⊆ V we denoteby G [ U ] the subgraph of G induced by the set U . For n ∈ N + we write [ n ] = { , . . . , n } . Wesay that V . ∪ V . ∪ · · · . ∪ V k = V is a balanced k -partition of V if for all i, j ∈ [ k ] it holds that | V i | ≤ | V j | + 1 . All logarithms are natural (base e ) if not specified otherwise. Probability and Erd˝os-R´enyi random graphs
We often denote random variables in boldfaceand write X ∼ D to denote that X is sampled from the distribution D . A p -biased coin , or a Bernoulli variable , is the outcome of a coin flip that yields with probability p and with probability − p . We use the special case of Markov’s inequality saying that if X is non-negative, then Pr[ X ≥ ≤ E [ X ] . We also need the following special case of the multiplicative Chernoff bound: if X is4 binomial random variable (i.e., the sum of i.i.d. Bernoulli variables) with expectation µ = E [ X ] ,then Pr[ X ≤ µ/ ≤ e − µ/ .We consider the Erd˝os-R´enyi distribution G ( n, p ) of random graphs on a fixed set V of n ver-tices. A random graph sampled from G ( n, p ) is produced by placing each potential edge { u, v } independently with probability p , ≤ p ≤ (the edge probability p may be a function of n ).A property of graphs is said to hold asymptotically almost surely on G ( n, p ( n )) if it holds withprobability that approaches as n approaches infinity.For a positive integer k , let X k be the random variable that counts the number of k -cliques in arandom graph from G ( n, p ) . It follows from Markov’s inequality that asymptotically almost surelythere are no k -cliques in G ( n, p ) whenever p and k are such that E [ X k ] = p ( k ) (cid:0) nk (cid:1) approaches as n approaches infinity. This is the case, for example, if p = n − η/ ( k − for k ≥ and η > . Actually,the clique number, i.e. the size of the largest clique, ω ( G ) for a graph G sampled from G ( n, p ) is awell studied quantity and very strong concentrations bounds are known for it. For instance, one ofthe first concentration results is that ω ( G ) = (2 − o (1)) log p ( n ) with probability as n → ∞ (seefor instance [BE76]). CNF formulas and resolution A literal over a Boolean variable x is either the variable x itself(a positive literal ) or its negation ¬ x (a negative literal ). A clause C = ℓ ∨ · · · ∨ ℓ w is a disjunctionof literals; we say that the width of C is w . The empty clause will be denoted by ⊥ . A CNF formula F = C ∧ · · · ∧ C m is a conjunction of clauses. We think of clauses as sets of literals and of CNFformulas as sets of clauses, so that order is irrelevant and there are no repetitions. For a formula F we denote by Vars ( F ) the set of variables of F .A resolution derivation from a CNF formula F is as an ordered sequence of clauses π =( D , . . . , D L ) such that for each i ∈ [ L ] either D i is a clause in F or there exist j < i and k < i such that D i is derived from D j and D k by the resolution rule B ∨ x C ∨ ¬ xB ∨ C , (2.1) D i = B ∨ C, D j = B ∨ x, D k = C ∨ ¬ x . We refer to B ∨ C as the resolvent of B ∨ x and C ∨ ¬ x over x , and to x as the resolved variable . The length (or size ) of a resolution derivation π = ( D , . . . , D L ) is L and it is denoted by | π | . A resolution refutation of F , or resolution proof for (the unsatisfiability of) F , is a resolution derivation from F that ends in the empty clause ⊥ .A resolution derivation π = ( D , . . . , D L ) can also be viewed as a labelled DAG with the set ofnodes { , . . . , L } and edges ( j, i ) , ( k, i ) for each application of the resolution rule deriving D i from D j and D k . Each node i in this DAG is labelled by its associated clause D i , and each non-sourcenode is also labelled by the resolved variable in its associated derivation step in the refutation. Aresolution refutation is called regular if along any source-to-sink path in its associated DAG everyvariable is resolved at most once.For a partial assignment ρ we say that a clause C restricted by ρ , denoted C ↾ ρ , is the trivial -clause if any of the literals in C is satisfied by ρ or otherwise is C with all falsified literals removed.We extend this definition to CNFs in the obvious way: ( C ∧ · · · ∧ C m ) ↾ ρ = C ↾ ρ ∧ · · · ∧ C m ↾ ρ .Applying a restriction preserves (regular) resolution derivations. To see this, observe that in every5pplication of the resolution rule, the restricted consequence either becomes identically 1, or it isobtained, as before, by resolving the two restricted premises, or it is a weakening of one of them,but weakenings can be removed at no cost. Thus, we have: Fact 2.1.
Let π be a (regular) resolution refutation of a CNF formula F . For any partial assignment ρ to the variables of F there is an efficiently constructible (regular) resolution refutation π ↾ ρ of theCNF formula F ↾ ρ , so that the length of π ↾ ρ is at most the length of π . Branching programs
A branching program on variables x , . . . , x n is a DAG that has one sourcenode and where every non-sink node is labelled by one of the variables x , . . . , x n and has exactlytwo outgoing edges labelled and . The size of a branching program is the total number of nodesin the graph. In a read-once branching program it holds in addition that along every path everyvariable appears as a node label at most once.For each node a in a branching program, let X ( a ) denote the variable that labels a , and let a and a be the nodes that are reached from a through the edges labelled and , respectively. Atruth-value assignment σ : { x , . . . , x n } → { , } determines a path in a branching program in thefollowing way. The path starts at the source node. At an internal node a , the path is extended alongthe edge labelled σ ( X ( a )) so that the next node in the path is a σ ( X ( a )) . The path ends when it reachesa sink. We write path( σ ) for the path determined by σ . When extending the path from a node a tothe node a σ ( X ( a )) , we say that the answer to the query X ( a ) at a is σ ( X ( a )) and that the path sets the variable X ( a ) to the value σ ( X ( a )) . For a node a of path( σ ) , let β ( σ, a ) be the restriction of σ to the variables that are queried in path( σ ) in the segment of the path that goes from the sourceto a . For each node a of the branching program, let β ( a ) be the maximal partial assignment thatis contained in every β ( σ, a ) for all σ such that path( σ ) passes through a . Equivalently, this is theset of all those assignments x i γ for which the query x i is made, and answered by γ , alongevery consistent path from the source to a . If the program is read-once, the consistency conditionbecomes redundant.The falsified clause search problem for an unsatisfiable CNF formula F is the task of findinga clause C ∈ F that is falsified by a given truth value assignment σ . A branching program P onthe variables Vars ( F ) solves the falsified clause search problem for F if each sink is labelled bya clause of F such that for every assignment σ , the clause that labels the sink reached by path( σ ) is falsified by σ . The minimal size of any regular resolution refutation of an unsatisfiable CNFformula F is exactly the same as the minimal size of any read-once branching program solving thefalsified clause search problem for F . This can be seen by taking the refutation DAG and reversingthe edges to get a branching program or vice versa. For a formal proof see, e.g., [Kra95, Theorem4.3]. The k -clique formula In order to analyse the complexity of resolution proofs that establish that agiven graph does not contain a k -clique we must formulate the problem as a propositional formulain conjunctive normal form (CNF). We consider two distinct encodings for the clique problemoriginally defined in [BIS07].The first propositional encoding we present, Clique ( G, k ) , is based on mapping of vertices toclique members. This formula is defined over variables x v,i ( v ∈ V, i ∈ [ k ]) and consists of the6ollowing set of clauses: ¬ x u,i ∨ ¬ x v,j i, j ∈ [ k ] , i = j, u, v ∈ V, { u, v } / ∈ E , (2.2a) _ v ∈ V x v,i i ∈ [ k ] , (2.2b) ¬ x u,i ∨ ¬ x v,i i ∈ [ k ] , u, v ∈ V, u = v , (2.2c)We refer to (2.2a) as edge axioms , (2.2b) as clique axioms and (2.2c) as functionality axioms . Notethat Clique ( G, k ) is satisfiable if and only if G contains a k -clique, and that this is true even ifclauses (2.2c) are omitted—we write Clique ∗ ( G, k ) to denote this formula with only clauses (2.2a)and (2.2b).The second version of clique formulas that we consider is the block encoding Clique block ( G, k ) .This formula differs from the previous ones in that it requires a k -clique that has a certain “block-respecting” structure. Let V ˙ ∪ V ˙ ∪ · · · ˙ ∪ V k = V be a balanced k -partition of V , that is a partitionof V into k disjoint sets each of them of size at most one integer away from | V | k . The formula Clique block ( G, k ) , defined over variables x v , encodes the fact that the graph contains a transversal k -clique , that is, a k -clique in which each clique member belongs to a different block. Formally,for any positive k and any graph G , the formula Clique block ( G, k ) consists of the following set ofclauses: ¬ x u ∨ ¬ x v u, v ∈ V, u = v, { u, v } / ∈ E , (2.3a) _ v ∈ V i x v i ∈ [ k ] , (2.3b) ¬ x u ∨ ¬ x v i ∈ [ k ] , u, v ∈ V i , u = v . (2.3c)We refer to (2.3a) as edge axioms , (2.3b) as clique axioms , and (2.3c) as functionality axioms .Note that a graph can contain a k -clique but contain no transversal k -clique for a given partition.Intuitively it is clear that proving that a graph does not contain a transversal k -clique should be easierthan proving it does not contain any k -clique, since any proof of the latter fact must in particularestablish the former. We make this intuition formal below. Lemma 2.2 ([BIS07]) . For any graph G and any k ∈ N + , the size of a minimum regular resolu-tion refutation of Clique ( G, k ) is bounded from below by the size of a minimum regular resolutionrefutation of Clique block ( G, k ) . This lemma was proven in [BIS07] for tree-like and for general resolution via a restriction argu-ment, and it is straightforward to see that the same proof holds for regular resolution as well.
Before proving our main n Ω( k ) lower bound, in this section we exhibit classes of graphs whoseclique formulas have regular resolution refutations of fixed-parameter tractable length, i.e., length7 ( k ) · n O (1) for some function f . This illustrates the strength of regular resolution for the k -cliqueproblem. We note that the upper bounds claimed in this section hold not only for Clique ( G, k ) buteven for the subformula Clique ∗ ( G, k ) that omits the functionality axioms (2.2c).The first example is the class of ( k − -colourable graphs. Such graphs are hard for tree-likeresolution [BGL13], and the known algorithms that distinguish them from graphs that contain k -cliques are highly non-trivial [Lov79, Knu94]. The second example is the class of graphs that havea homomorphism into a fixed k -clique free graph.Recall that a homomorphism from a graph G = ( V, E ) into a graph G ′ = ( V ′ , E ′ ) is a mapping h : V → V ′ that maps edges { u, v } ∈ E into edges { h ( u ) , h ( v ) } ∈ E ′ . A graph is ( k − -colourable if and only if it has a homomorphism into the ( k − -clique, which is of course k -clique free. Therefore our second example is a generalization of the first one (but the function f ( k ) becomes larger).Both upper bounds follows from a generic procedure, based on Algorithm 1, that builds read-once branching programs for the falsified clause search problem for Clique ∗ ( G, k ) .Given a k -clique free graph G define I ( G ) = (cid:8) G (cid:2) b N ( R ) (cid:3) : R is a clique in G (cid:9) . (3.1) Proposition 3.1.
There is an efficiently constructible read-once branching program for the falsifiedclause search problem on formula Clique ∗ ( G, k ) of size at most | I ( G ) | · k · | V ( G ) | .Proof. We build the branching program recursively, following the strategy laid out by Algorithm 1.For the base case k = 1 , G must be the graph with no vertices. The branching program is a singlesink node that outputs the clique axiom of index , i.e., the empty clause.For k > , fix n = | V ( G ) | and an ordering v , . . . , v n of the vertices in V ( G ) . We first build adecision tree T by querying the variables x v ,k , x v ,k , . . . in order, until we get an answer , or untilall variables with second index k have been queried. If x v j ,k = 0 for all j ∈ [ n ] then the k th cliqueaxiom (2.2b) is falsified by the assignment (see line 9). Otherwise, let v be the first vertex in theorder where x v,k = 1 . The decision tree now queries x w,i for all w ∈ V ( G ) \ N ( v ) and all i < k tocheck whether an edge axiom involving v is falsified (lines 4–5). If any of these variables is set to the branching stops and the leaf node is labelled with the corresponding edge axiom ¬ x v,k ∨ ¬ x w,i .The decision tree T built so far has at most kn nodes, and we can identify n “open” leaf nodes a v , a v , . . . , a v n , where a v i is the leaf node reached by the path that sets x v i ,k = 1 and that doesnot yet determine the answer to the search problem. Let us focus on a specific node a v for some v ∈ V ( G ) . The partial assignment path( a v ) sets v to be the k th member of the clique and everyvertex in V ( G ) \ N ( v ) to not be in the clique. Let G v be the subgraph induced on G by N ( v ) , let S v be the set of variables x w,i for w ∈ N ( v ) and i < k , and let ρ v be the partial assignment setting x w,i = 0 for w ∈ V ( G ) \ N ( v ) and i < k . Clearly ρ v ⊆ path( a v ) .By the inductive hypothesis there exists a branching program B v that solves the search problemon Clique ∗ ( G v , k − querying only variables in S v . This corresponds to the recursive call for thesubgraph G v and k − (lines 6–8). If we attach each B v to a v we get a complete branching programfor Clique ∗ ( G, k ) . This is read-once because B v only queries variables in S v and these variablesare not in path( a v ) . 8 lgorithm 1 Read-once branching program for the falsified clause search problem on
Clique ∗ ( G, k ) . Input : k ∈ N + , a k -clique free graph G , an assignment α : { x v,i for v ∈ V ( G ) , i ∈ [ k ] } → { , } Output :
A clause of
Clique ∗ ( G, k ) falsified by α Search ( G, k, α ) : begin for v ∈ V ( G ) do if α ( x v,k ) = 1 then for w ∈ V ( G ) \ N ( v ) and i < k do if α ( x w,i ) = 1 then return edge axiom ¬ x v,k ∨ ¬ x w,i (2.2a) G ′ ← G [ N ( v )] α ′ ← α restricted to variables x w,j for w ∈ V ( G ′ ) and ≤ j ≤ k − return Search ( G ′ , k − , α ′ ) return the k th clique axiom (2.2b)To prove that the composed program is correct we consider an assignment σ to the variablesin S v and show that the clause output by B v on σ is also a valid output for the search problem on Clique ∗ ( G, k ) , i.e., it is falsified by the assignment path( a v ) ∪ σ . Actually we show the strongerclaim that it is falsified by ρ v ∪ σ , which is a subset of path( a v ) ∪ σ . To this end, note that if theoutput of B v on σ is an edge axiom of Clique ∗ ( G v , k − , this must be some ¬ x u,i ∨ ¬ x w,j for i, j < k , which is also an edge axiom of Clique ∗ ( G, k ) and is falsified by σ . Now if the output of B v on σ is the i th clique axiom of Clique ∗ ( G v , k − , then σ falsifies W w ∈ N ( v ) x v,i , and therefore ρ v ∪ σ falsifies the i th clique axiom of Clique ∗ ( G, k ) .The construction so far is correct but produces a very large branching program (in particular ithas tree-like structure on top). In order to create a smaller branching program, we observe that if u, v ∈ V ( G ) are such that N ( u ) = N ( w ) then G u = G w , B u = B w and ρ u = ρ w . This observationallows us to merge together all nodes a v that have the same value of N ( v ) into a single node, and toidentify all the corresponding copies of the same branching program B v . Now let us focus on somenode a ∗ obtained by this merge process, and pick arbitrarily some a v that was merged into it (thespecific choice is irrelevant). By construction ρ v is consistent with all paths reaching a ∗ , but we canclaim further: ρ v is consistent with all paths passing through a ∗ because B v only queries variablesin S v , which is disjoint from the domain of ρ v . Because of this last fact all paths that pass throughnode a ∗ and reach an output node b ∗ in the attached copy of B v must contain the partial assignment ρ v ∪ σ , where σ is the common partial assignment consistent with all paths from the root of B v to b ∗ . If b ∗ outputs an edge axiom, this is already falsified by σ because of the correctness of B v . If b ∗ outputs the i th clique axiom, the correctness of B v guarantees that σ falsifies the i th axiom for G v , and therefore ρ v ∪ σ falsifies the i th clique axiom of G . Hence the new branching program iscorrect.This merge process leads to having only one subprogram for each distinct induced subgraph at9ach level of the recursion. In order to bound the size of this program, we decompose it into k levels.The source is at level zero and corresponds to the graph G . At level i there are nodes correspondingto all subgraphs induced by the common neighbourhood of cliques of size i . Each node in the i thlevel connects to the nodes of the ( i + 1) th level by a branching program of size at most kn . Noticethat an induced subgraph in I ( G ) cannot occur twice in the same layers, so the total size of the finalbranching program is at most | I ( G ) | · k n nodes.We now proceed to prove the upper bounds mentioned previously. A graph G that has a homo-morphism into a small k -clique free graph H may still have a large set I ( G ) , making Proposition 3.1inefficient. The first key observation is that if G has a homomorphism into a graph H then it is asubgraph of a blown up version of H , namely, of a graph obtained by transforming each vertex of H into a “cloud” of vertices where a cloud does not contain any edge, two clouds corresponding totwo adjacent vertices in H have all possible edges between them, and two clouds corresponding totwo non-adjacent vertices in H have no edges between them. A second crucial point is that if G ′ isa blown up version of H then it turns out that | I ( G ′ ) | = | I ( H ) | , making Proposition 3.1 effectivefor G ′ . The upper bound then follows from observing that the task of proving that G is k -cliquefree should not be harder than the same task for a supergraph of G . Indeed Fact 3.2 formalises thisintuition. It is interesting to observe that the constructions in Proposition 3.1 and in Fact 3.2 areefficient. The non-constructive part is guessing the homomorphism to H . Fact 3.2.
Let G = ( V, E ) and G ′ = ( V ′ , E ′ ) be graphs with no k -clique such that V ⊆ V ′ and E ⊆ E ′ ∩ (cid:0) V (cid:1) . If Clique ∗ ( G ′ , k ) has a (regular) refutation of length L , then Clique ∗ ( G, k ) has a(regular) refutation of length at most L .Proof. Consider the partial assignment ρ that sets x v,i = 0 for every v V and i ∈ [ k ] . Therestricted formula Clique ∗ ( G ′ , k ) ↾ ρ is isomorphic to Clique ∗ ( e G, k ) , where V ( e G ) = V and E ( e G ) = E ′ ∩ (cid:0) V (cid:1) , and thus, by Fact 2.1, has a (regular) refutation π of length at most L . Removing edgesfrom a graph only introduces additional edge axioms (2.2a) in the corresponding formula, therefore Clique ∗ ( e G, k ) ⊆ Clique ∗ ( G, k ) and π is a valid refutation of Clique ∗ ( G, k ) as well.It was shown in [BGL13] that the k -clique formula of a complete ( k − -partite graph on n vertices has a regular resolution refutation of length k n O (1) , although the regularity is not stressedin that paper. Since it is instructive to see how this refutation is constructed in this framework, wegive a self-contained proof. Proposition 3.3 ([BGL13, Proposition 5.3]) . If G is a ( k − -colourable graph on n vertices, thenClique ∗ ( G, k ) has a regular resolution refutation of length at most k k n .Proof. Let V = V ( G ) and let V ˙ ∪ V ˙ ∪ . . . ˙ ∪ V ( k − be a partition of V into colour classes. Definethe graph G ′ = ( V, E ′ ) where the edge set E ′ has an edge between any pair of vertices belongingto two different colour classes. Clearly G is a subgraph of G ′ . Observe that any clique R in G ′ hasat most one vertex in each colour class, and that the common neighbours of R are all the verticesin the colour classes not touched by R .Therefore, there is a one-to-one correspondence between the members of I ( G ′ ) and the subsetsof [ k − . By Proposition 3.1 there is a read-once branching program for the falsified clause search10roblem on formula Clique ∗ ( G ′ , k ) of size at most k k n . This read-once branching programcorresponds to a regular resolution refutation of Clique ∗ ( G ′ , k ) of the same size. By Fact 3.2 theremust be a regular resolution refutation of size at most k k n for Clique ∗ ( G, k ) as well.Next we generalize Proposition 3.3 to graphs G that have a homomorphism to a k -clique freegraph H . Proposition 3.4. If G is a graph on n vertices that has a homomorphism into a k -clique free graph H on m vertices, then Clique ∗ ( G, k ) has a regular resolution refutation of length at most m k k n .Proof. Fix a homomorphism h : V ( G ) → V ( H ) and an ordering u , . . . , u m of the vertices of H .Let V ˙ ∪ V ˙ ∪ . . . ˙ ∪ V m be the partition of V ( G ) such that V i is the set of vertices of G mapped to u i by h . We define the graph G ′ = ( V, E ′ ) where E ′ = [ { u i ,u j }∈ E ( H ) V i × V j , (3.2)that is, G ′ is a blown up version of H that contains G as a subgraph. To prove our result we note that,by Proposition 3.1, there is a read-once branching program for the falsified clause search problemon Clique ∗ ( G ′ , k ) —and hence also a regular resolution refutations of the same formula—of sizeat most | I ( G ′ ) | · k n . This implies that, by Fact 3.2, there is a regular resolution refutation of Clique ∗ ( G, k ) of at most the same size.To conclude the proof it remains only to show that | I ( G ′ ) | ≤ m k . By construction, h mapsinjectively a clique R ⊆ V ( G ′ ) into a clique R H ⊆ V ( H ) of the same size. Moreover, note thatif U = b N ( R H ) , then b N ( R ) = ∪ u i ∈ U V i . Therefore, for any clique R ′ ⊆ V ( G ′ ) that is mappedby h to R H it holds that b N ( R ) = b N ( R ′ ) , i.e., b N ( R ′ ) is completely characterized by the cliquein H it is mapped to. Thus I ( G ) has at most one element for each clique in H and we have that | I ( G ′ ) | = | I ( H ) | . Finally, note that | I ( H ) | ≤ m k since, being k -clique free, H cannot have morethan P k − i =0 m i ≤ m k cliques. The main result of this paper is an average case lower bound of n Ω( k ) for regular resolution for the k -clique problem. As we saw in Section 2, the k -clique problem can be encoded in different waysand depending on the preferred formula the range of k for which we can obtain a lower bound differs.In this section we present a summary of our results for the different encodings. Theorem 4.1.
For any real constant ǫ > , any sufficiently large integer n , any positive integer k ≤ n / − ǫ , and any real ξ > , if G ∼ G ( n, n − ξ/ ( k − ) is an Erd˝os-R´enyi random graph, then, withprobability at least − exp( −√ n ) , any regular resolution refutation of Clique block ( G , k ) has lengthat least n Ω( k/ξ ) . The parameter ξ determines the density of the graph: the larger ξ the sparser the graph andthe problem of determining whether G contains a k -clique becomes easier. For constant ξ , the11dge probability implies the graph G has clique number concentrated around k/ξ and the theoremyields a n Ω( k ) lower bound which is tight up to the multiplicative constant in the exponent. Thelower bound decreases smoothly with the edge density and is non-trivial for ξ = o ( √ k ) .A problem which is closely related to the problem we consider is that of distinguishing a randomgraph sampled from G ( n, p ) from a random graph from the same distribution with a planted k -clique. The most studied setting is when p = 1 / . In this scenario the problem can be solved inpolynomial time with high probability for k ≈ √ n [Kuˇc95, AKS98]. It is still an open problemwhether there exists a polynomial time algorithm solving this problem for log n ≪ k ≪ √ n .For G ∼ G ( n, / , setting ξ = k −
12 log ( n ) , Theorem 4.1 implies that to refute Clique block ( G , k ) asymptotically almost surely regular resolution requires n Ω(log ( n ) /k ) size; which is n Ω(log n ) size for k = O (log n ) and super-polynomial size for k = o(log n ) . We note that, in the case k = O (log n ) ,the lower bound is tight. This follows from Proposition 3.1 since asymptotically almost surely thereare at most n O (log n ) different cliques in G ∼ G ( n, / (because asymptotically almost surely thelargest clique has size at most n ) and, therefore, the set I ( G ) in Proposition 3.1 has size atmost n O (log n ) .An interesting question is whether Theorem 4.1 holds for larger values of k . We show that forthe formula Clique ( G, k ) (recall that by Lemma 2.2 this encoding is easier for the purpose of lowerbounds) we can prove the lower bound for k ≤ n / − ǫ as long as the edge density of the graph isclose to the threshold for containing a k -clique. Theorem 4.2.
For any real constant ǫ > , any sufficiently large integer n , any positive integer k ,and any real ξ > such that k √ ξ ≤ n / − ǫ , if G ∼ G ( n, n − ξ/ ( k − ) is an Erd˝os-R´enyi ran-dom graph, then, with probability at least − exp( −√ n ) , any regular resolution refutation ofClique ( G , k ) has length at least n Ω( k/ξ ) . In this paper we prove Theorem 4.1 and we refer to the conference version of this paper [ABdR + In this section we define a combinatorial property of graphs, which we call clique-denseness , andprove that if a k -clique-free graph G is clique-dense with the appropriate parameters, then thisimplies a lower bound n Ω( k ) on the length of any regular resolution refutation of the k -clique formulaon G .In order to argue that regular resolution has a hard time certifying the k -clique-freeness of agraph G , one property that seems useful to have is that for every small enough clique in the graphthere are many ways of extending it to a larger clique. In other words, if R ⊆ V forms a cliqueand R is small, we would like the common neighbourhood b N V ( R ) to be large. This motivates thefollowing definitions. 12 efinition 5.1 (Neighbour-dense set) . Given G = ( V, E ) and q, r ∈ R + , a set W ⊆ V is q -neigh-bour-dense for R ⊆ V if (cid:12)(cid:12) b N W ( R ) (cid:12)(cid:12) ≥ q . We say that W is ( r, q ) -neighbour-dense if it is q -neigh-bour-dense for every R ⊆ V of size | R | ≤ r . If W is an ( r, q ) -neighbour-dense set, then we know that any clique of size r can be extendedto a clique of size r + 1 in at least q different ways by adding some vertex of W . Note, however,that the definition of ( r, q ) -neighbour-dense is more general than this since R is not required to bea clique.Next we define a more robust notion of neighbour-denseness. For some settings of r and q ofinterest to us it is too much to hope for a set W that is q -neighbour-dense for every R ⊆ V of sizeat most r . In this case we would still like to be able to find a “mostly neighbour-dense” set W in thesense that we can “localize” bad (i.e., those for which W fails to be q -neighbour-dense) sets R ⊆ V of size | R | ≤ r . Definition 5.2 (Mostly neighbour-dense set) . Given G = ( V, E ) and r ′ , r, q, s ∈ R + with r ′ ≥ r , aset W ⊆ V is ( r ′ , r, q, s ) -mostly neighbour-dense if there exists a set S ⊆ V of size | S | ≤ s suchthat for every R ⊆ V with | R | ≤ r ′ for which W is not q -neighbour-dense, it holds that | R ∩ S | ≥ r . In what follows, it might be helpful for the reader to think of r ′ and r as linear in k , and q and s as polynomial in n , where we also have s ≪ q .Now we are ready to define a property of graphs that makes it hard for regular resolution tocertify that graphs with this property are indeed k -clique-free. Definition 5.3 (Clique-dense graph) . Given k ∈ N + and t, s, ε ∈ R + , ≤ t ≤ k we say that agraph G = ( V, E ) with a k -partition V ∪ · · · ∪ V k = V is ( k, t, s, ε ) -clique-dense if there exist r, q ∈ R + , r ≥ k/t , such that1. V i is ( tr, tq ) -neighbour-dense for all i ∈ [ k ] , and2. every ( r, q ) -neighbour-dense set W ⊆ V is ( tr, r, q ′ , s ) -mostly neighbour-dense for q ′ = εrs ε log s .Remark ( k − -partite graph is not clique-dense) . Since the property of clique-denseness in Definition 5.3 is a sufficient condition for the lower bound, it is worth to pause andobserve that this property does not hold for examples such as ( k − -colourable graphs, whichhave non-trivially short proofs.Indeed, consider the ( k − -colourable graph G = ( V, E ) with balanced colour classes andmaximum edge set. Namely, V = S c U c for c ∈ [ k − and | U c | = n/ ( k − , and the edges of G are all pairs { u, v } for u ∈ U c and v ∈ U c ′ with c = c ′ . The graph G satisfies property (1) ofclique-denseness for any k -partition of V that splits each colour class roughly equally among parts,but fails to satisfy property (2) in a rather extreme way. To see why, fix any integer r < k − andlet W be the union of r + 1 arbitrarily chosen colour classes. The set W is ( r, q ) -neighbour-densefor any q up to n/ ( k − , because the common neighbourhood of any r vertices in V must containone of the colour classes U c ⊆ W .Can W be ( tr, r, q ′ , s ) -mostly neighbour-dense for some choice of parameters? First note that tr ≥ r + 1 (since r ≤ k implies t ≥ ) and that b N W ( R ) = ∅ for any set R of size r + 1 that13as one vertex from each colour class in W . So in order for W to be ( tr, r, q ′ , s ) -mostly neighbour-dense there should be a set S of size s ≪ q ′ ≤ n/ ( k − that has a large intersection with anysuch R . This, however, is not possible since S cannot completely cover any of the colour classesin W (because s ≪ n/ ( k − ) and thus, for any choice of S , there are sets R completely disjointfrom S for which b N W ( R ) = ∅ . Theorem 5.4.
Given k ∈ N + and t, s, ε ∈ R + if the graph G = ( V, E ) with balanced k -partition V ∪ · · · ∪ V k = V is ( k, t, s, ε ) -clique-dense, then every regular resolution refutation of the CNFformula Clique block ( G, k ) has length at least Ω (cid:0) s εk/t (cid:1) . The value of q ′ in Definition 5.3 can be tailored in order to prove Theorem 4.1 for slightly largervalues of k . For example, setting q ′ = 3 εs ε log s and making the necessary modifications in theproof would yield Theorem 4.1 for k ≪ n / but for a smaller range of edge densities. A similaradjustment was done in the conference version of this paper [ABdR +
18] to obtain Theorem 4.2 for k ≪ n / .We will spend the rest of this section establishing Theorem 5.4. Fix r, q ∈ R + witnessing that G is ( k, t, s, ε ) -clique-dense as per Definition 5.3. We first note that we can assume that tr ≤ k sinceotherwise, by property 1 of Definition 5.3, G contains a block-respecting k -clique and the theoremfollows immediately.By the discussion in Section 2 it is sufficient to consider read-once branching programs, sincethey are equivalent to regular resolution refutations, and so in what follows this is the language inwhich we will phrase our lower bound. Thus, for the rest of this section let P be an arbitrary, fixedread-once branching program that solves the falsified clause search problem for Clique block ( G, k ) .We will use the convention of referring to “vertices” of the graph G and “nodes” of the branchingprogram P to distinguish between the two. We sometime abuse notation and say that a vertex v ∈ V is set to or to when we mean that the corresponding variable x v is set to or to .Recall that for a node a of P , β ( a ) denotes the maximal partial assignment that is contained inevery β ( σ, a ) for all σ such that path( σ ) passes through a , where β ( σ, a ) is the restriction of σ tothe variables that are queried in path( σ ) in the segment of the path that goes from the source to a . For any partial assignment β we write β to denote the partial assignment that contains exactlythe variables that are set to in β . Clearly, if β falsifies an edge axiom or a functionality axiom,then so does β . Furthermore, for any γ ⊇ β , if β falsifies an axiom so does γ . We will use thismonotonicity property of partial assignments throughout the proof.For each node a of P and each index i ∈ [ k ] we define two sets of vertices V i ( a ) = { u ∈ V i | β ( a ) sets x u to } (5.1a) V i ( a ) = { u ∈ V i | β ( a ) sets x u to } (5.1b)of G . Observe that for β = β ( a ) the set of vertices referenced by variables in β is S i V i ( a ) .Intuitively, one can think of V i ( a ) and V i ( a ) as the only sets of vertices in V i assigned and ,respectively, that are “remembered” at the node a (in the language of resolution, they correspond tonegative and positive occurrences of variables in the clause D a associated with the node a ). Otherassignments to vertices in V i encountered along some path to a have been “forgotten” and may notbe queried any more on any path starting at a . Formally, we say that a vertex v is forgotten at a
14f there is a path from the source of P to a passing through a node b where v is queried, but v isnot in V i ( a ) nor in V i ( a ) . Furthermore, we say index i is forgotten at a if some vertex v ∈ V i is forgotten at a . Of utter importance is the fact that these notions are persistent: if a variable oran index is forgotten at a node a , then it will also be the case for any node reachable from a by apath. We say that a path in P ends in the i th clique axiom if the clause that labels its last node isthe clique axiom (2.3b) of Clique block ( G, k ) with index i . The above observation implies that theindex i cannot be forgotten at any node along such a path.We establish our lower bound via a bottleneck counting argument for paths in P . To this end,let us define a distribution D over paths in P by the following random process. The path starts atthe source and ends whenever it reaches a sink of P . At an internal node a with successor nodes a and a , reached by edges labelled and respectively, the process proceeds as follows.1. If X ( a ) = x u for u ∈ V i and i is forgotten at a then the path proceeds via the edge labelled to a .2. If X ( a ) = x u and β ( a ) ∪ { x u = 1 } falsifies an edge axiom (2.3a) or a functionality ax-iom (2.3c), then the path proceeds to a .3. Otherwise, an independent s − (1+ ε ) -biased coin is tossed with outcome γ ∈ { , } and therandom path proceeds to a γ .We say that in cases 1 and 2 the answer to the query X ( a ) is forced . Note that any path α in thesupport of D must end in a clique axiom since α does not falsify any edge or functionality axiom byitem 2 in the construction. Moreover, a property that will be absolutely crucial is that only answers can be forced—answers are always the result of a coin flip. Claim 5.5.
Every path in the support of D sets at most k variables to .Proof. Let α be a path in the support of D . We argue that for each i ∈ [ k ] at most one vertex u ∈ V i is such that the variable x u is set to on α . Let a and b be two nodes that appear in this order in α . If for some i ∈ [ k ] , and for some u, v ∈ V i , x u is set to by α at node a and x v is queried at b ,then v = u by regularity and, by definition of D , the answer to query x v will be forced to , eitherto avoid violating a functionality or an edge axiom, or because i is forgotten at b .Let us call a pair ( a, b ) of nodes of P useful if there exists an index i such that V i ( b ) = ∅ , i is notforgotten at b , and the set V i ( b ) \ V i ( a ) is ( r, q ) -neighbour-dense. In particular, if a appears before b in some path, then V i ( a ) = ∅ and V i ( a ) ⊆ V i ( b ) . For each useful pair ( a, b ) , let i ( a, b ) be anarbitrary but fixed index witnessing that ( a, b ) is useful. A path is said to usefully traverse a usefulpair ( a, b ) if it goes through a and b in that order and sets at most ⌈ k/t ⌉ variables to between a and b (with a included and b excluded).As already mentioned, the proof of Theorem 5.4 is based on a bottleneck counting argumentin the spirit of [Hak85], with the twist that we consider pairs of bottleneck nodes. To establish thetheorem we make use of the following two lemmas which will be proven subsequently. Lemma 5.6.
Every path in the support of D usefully traverses a useful pair. emma 5.7. For every useful pair ( a, b ) , the probability that a random α chosen from D usefullytraverses ( a, b ) is at most s − εr/ . Combining the above lemmas, it is immediate to prove Theorem 5.4. By Lemma 5.6 the proba-bility that a random path α sampled from D usefully traverses some useful pair is . By Lemma 5.7,for any fixed useful pair ( a, b ) , the probability that a random α usefully traverses ( a, b ) is at most s − εr/ . By a standard union bound argument, it follows that the number of useful pairs is at least s εr/ , so the number of nodes in P cannot be smaller than Ω (cid:0) s εr/ (cid:1) ≥ Ω (cid:0) s εk/t (cid:1) (recall that r ≥ k/t according to Definition 5.3).To conclude the proof it remains only to establish Lemmas 5.6 and 5.7. Proof of Lemma 5.6.
Consider any path in the support of D . As we already remarked, this pathends in the i ∗ th clique axiom for some i ∗ ∈ [ k ] which in particular implies that V i ∗ ( b ) = ∅ andthat i ∗ is not forgotten at any b along this path. By Claim 5.5, the path sets at most k variables to and hence we can split it into t pieces by nodes a , a , . . . , a t ( a is the source, a t the sink) so thatbetween a j and a j +1 at most ⌈ k/t ⌉ variables are set to 1. It remains to prove that for at least one j ∈ [ t ] the set W j = V i ∗ ( a j ) \ V i ∗ ( a j − ) (5.2)is ( r, q ) -neighbour-dense. Note that this will prove Lemma 5.6 since by construction ( a j − , a j ) isthen a pair that is usefully traversed by the path.Towards contradiction, assume instead that no W j is ( r, q ) -neighbour-dense, i.e., that for all j ∈ [ t ] there exists a set of vertices R j ⊆ V with | R j | ≤ r such that (cid:12)(cid:12) b N W j ( R j ) (cid:12)(cid:12) ≤ q . Let R = S j ∈ [ t ] R j . Since the path ends in the i ∗ th clique axiom we have V i ∗ ( a t ) = V i ∗ . It follows thatthe sets W , . . . , W t in (5.2) form a partition of V i ∗ , and therefore (cid:12)(cid:12) b N V i ∗ ( R ) (cid:12)(cid:12) = X j ∈ [ t ] (cid:12)(cid:12) b N W j ( R ) (cid:12)(cid:12) ≤ X j ∈ [ t ] (cid:12)(cid:12) b N W j ( R j ) (cid:12)(cid:12) ≤ tq . (5.3)Since | R | ≤ P j ∈ [ t ] | R j | ≤ tr this contradicts the assumption that V i ∗ is ( tr, tq ) -neighbour-dense.Lemma 5.6 follows. Proof of Lemma 5.7.
Fix a useful pair ( a, b ) . Let E denote the event that a random path sampledfrom D usefully traverses ( a, b ) . Let i ∗ = i ( a, b ) , V ( a ) = S j ∈ [ k ] V j ( a ) , and W = V i ∗ ( b ) \ V i ∗ ( a ) .Notice that W is guaranteed to be ( r, q ) -neighbour-dense by our definition of i ( a, b ) . Since G is ( k, t, s, ε ) -clique-dense by assumption, this implies that W is ( tr, r, q ′ , s ) -mostly neighbour-dense,and we let S be the set that witnesses this as per Definition 5.2. We bound the probability of theevent E by a case analysis based on the size of the set V ( a ) . We remark that all probabilities in thecalculations that follow are over the choice of α ∼ D . Case 1 ( | V ( a ) | > r/ ) : In this case, we simply prove that already the probability of reaching a is small. By definition of V ( a ) , we have that | β ( a ) | = | V ( a ) | . Recall that every answer isnecessarily the result of a s − (1+ ε ) -biased coin flip, and that all these decisions are irreversible. Thatis, if a path ever decides to set a variable in V ( a ) to 0, then its case is lost and it is guaranteed tomiss a . Thus we can upper bound the probability of the event E by the probability that a random16 passes through a , and, in particular, by the probability of setting all variables in β ( a ) to asfollows: Pr[ E ] ≤ Pr[ α passes through a ] (5.4) ≤ (cid:0) s − (1+ ε ) (cid:1) | β ( a ) | (5.5) ≤ s − ε | V ( a ) | (5.6) ≤ s − εr/ . (5.7) Case 2 ( | V ( a ) | ≤ r/ ) : For every path α , let R ( α ) denote the set of vertices u set to by thepath α at some node between a and b (with a included and b excluded); note that R ( α ) = ∅ if α does not go through a and b , and that | R ( α ) | ≤ ⌈ k/t ⌉ for all paths α that satisfy the event E . Forthe sets R = { R : | R | ≤ ⌈ k/t ⌉ and (cid:12)(cid:12) b N W ( R ∪ V ( a )) (cid:12)(cid:12) < q ′ } (5.8a) R = { R : | R | ≤ ⌈ k/t ⌉ and (cid:12)(cid:12) b N W ( R ∪ V ( a )) (cid:12)(cid:12) ≥ q ′ } (5.8b)we have that Pr[ E ] = Pr[ E and R ( α ) ∈ R ] + Pr[ E and R ( α ) ∈ R ] . (5.9)The first term in (5.9) is bounded from above by the probability of R ( α ) ∈ R . Note that | R | ≤ ⌈ k/t ⌉ ≤ k/t ≤ tr/ (since r ≥ k/t ) for R ∈ R . Hence we have | R ∪ V ( a ) | ≤ tr/ r/ ≤ tr and therefore | ( R ∪ V ( a )) ∩ S | ≥ r by the choice of S . Thus, the probability of R ( α ) ∈ R is bounded by the probability that | R ( α ) ∩ S | ≥ r/ since | V ( a ) | ≤ r/ . But since S is small, we can now apply the union bound and conclude that Pr[ E and R ( α ) ∈ R ] ≤ Pr[ R ( α ) ∈ R ] (5.10) ≤ Pr[ | R ( α ) ∩ S | ≥ r/ (5.11) ≤ (cid:18) | S | r/ (cid:19) ( s − (1+ ε ) ) r/ (5.12) ≤ | S | r/ s − (1+ ε ) r/ (5.13) ≤ s − εr/ , (5.14)where for (5.12) we used the same “irreversibility” argument as in Case 1.We now bound the second term in (5.9). First note that, by definition of W , if α is a path thatpasses through a and b in this order, then all u ∈ W must be set to in α at some node between a and b . For each path in the support of D that passes through a and b , some of the vertices in W will be set to zero as a result of a coin flip and others will be forced choices.Fix a path α contributing to the second term in (5.9). We claim that along this path all the ≥ q ′ variables in b N W ( R ( α ) ∪ V ( a )) are set to as a result of a coin flip. Indeed, since V i ∗ ( b ) = ∅ and i ∗ is not forgotten at b , by the monotonicity property the same holds for every node along α before b .This implies that the answer to a query of the form x u ( u ∈ W ) made along α cannot be forced by17either item 1 (forgetfulness) in the definition of D nor by a functionality axiom. Moreover, since V ( c ) ⊆ R ( α ) ∪ V ( a ) for any node c on the path α between a and b , it holds that all variables x u with u ∈ b N W ( R ( α ) ∪ V ( a )) can not be forced to by an edge axiom either.The analysis of the second term in (5.9) is completed by the same type of argument as in Case1, where we again use the fact that, due to the read-once property of the branching program, thedecisions that the random path makes are irreversible: Pr[ E and R ( α ) ∈ R ] ≤ Pr[ α flips ≥ q ′ coins and gets 0-answers ] (5.15) ≤ (1 − s − (1+ ε ) ) q ′ (5.16) ≤ s − εr/ . (5.17)Adding (5.14) and (5.17) we obtain the lemma. In this section we show that asymptotically almost surely an Erd˝os-R´enyi random graph G ∼ G ( n, p ) is ( k, t, s, ε ) -clique-dense for the right choice of parameters. Theorem 6.1.
For any real constant ε ∈ (0 , / , any sufficiently large integer n , any positiveinteger k ≤ n / − ε , and any real ξ > , if G ∼ G ( n, n − ξ/ ( k − ) is an Erd˝os-R´enyi randomgraph then with probability at least − exp( −√ n ) it holds that G is ( k, t, s, ε ) -clique-dense with t = 32 ξ/ε and s = √ n . As a corollary of Theorem 5.4 and Theorem 6.1 we obtain Theorem 4.1, the main result of thispaper.
Proof of Theorem 4.1.
Clearly t ≥ as required by Definition 5.3. We can also assume w.l.o.g.that t ≤ k since otherwise k/ξ ≤ / ( ξǫ ) ≤ O (1) and the bound becomes trivial. By plugging inthe parameters given by Theorem 6.1 to Theorem 5.4 we immediately get that any regular refutation π of Clique block ( G , k ) has length | π | ≥ Ω (cid:0) s εk/t (cid:1) ≥ n Ω( k/ξ ) , (6.1)as stated.We will spend the rest of this section proving Theorem 6.1.Let δ = 2 ξ/ ( k − . We show that, with probability at least − e −√ n , the random graph G is ( k, t, s, ε ) -clique-dense for parameters as in the statement of the theorem, r = 4 k/t and q = n − tδr kt .18ecall that q ′ = εrs ε log s . Let us argue that the parameters we use satisfy constraints tδr ≤ ε , (6.2) log k + tr log n ≤ n − tδr k · nn / , (6.3) qn − tδr s tr ≥ n ε , (6.4) q ′ ≤ qn − tδr · log nn ε/ , (6.5) tr ≤ q , (6.6)which will be used further on in the proof.As a first step note that tδr = 8 ξkt ( k − ≤ ε , (6.7)and hence (6.2) holds. Equation (6.3) follows from the chain of inequalities log k + tr log n ≤ tr log n = 8 k log nt ≤ k log n ≤ n / − ε log n k ≤ n − tδr k · nn / . (6.8)To obtain (6.4) observe that qn − tδr s tr = n − tδr +1 / k ≥ n − tδr +2 ε ≥ n ε . (6.9)To see that (6.5) holds, note that q ′ = 2 εkn (1+ ε ) / log nt ≤ k n (1+ ε ) / log n kt ≤ n − ε/ log n kt ≤ qn − tδr · log nn ε/ . (6.10)Finally, for (6.6), we just observe that tr = 4 kt ≤ k k ≤ n − tδr kt = q , (6.11)using the fact that k ≥ t and k ≤ n − tδr .We must now prove that asymptotically almost surely G is ( k, t, s, ε ) -clique-dense for the cho-sen parameters. All probabilities in this section are over the choice of G , and all previously intro-duced concepts like b N W ( R ) , neighbour-denseness etc. should be understood with respect to G aswell (so that they are actually random variables and events in this sample space). Let V = V ( G ) and V ∪ · · · ∪ V k = V be a balanced k -partition of V .The fact that asymptotically almost surely V i is ( tr, tq ) -neighbour-dense for all i ∈ [ k ] is quiteimmediate. First, for any i ∈ [ k ] and any R ⊆ V with | R | ≤ tr , E (cid:2)(cid:12)(cid:12) b N V i ( R ) (cid:12)(cid:12)(cid:3) = | V i \ R | n − δ | R | ≥ (cid:16) nk − tr (cid:17) n − δtr ≥ (cid:16) nk − q (cid:17) n − δtr ≥ n − δtr k , (6.12)19here the second-to-last inequality follows from (6.6) and the last inequality from the trivial factthat q ≤ nk . Hence, we can bound the probability that there exists an i ∈ [ k ] such that V i is not ( tr, tq ) -neighbour-dense by Pr h ∃ i ∈ [ k ] ∃ R ⊆ V, | R | = ⌊ tr ⌋ ∧ (cid:12)(cid:12) b N V i ( R ) (cid:12)(cid:12) ≤ tq i (6.13) ≤ k (cid:18) ntr (cid:19) max i,R Pr h(cid:12)(cid:12) b N V i ( R ) (cid:12)(cid:12) ≤ tq i (6.14) ≤ kn tr max i,R Pr (cid:20)(cid:12)(cid:12) b N V i ( R ) (cid:12)(cid:12) ≤ n − tδr k (cid:21) (6.15) ≤ kn tr exp (cid:18) − n − tδr k (cid:19) (6.16) ≤ exp (cid:18) − n − tδr k · (cid:18) − nn / (cid:19)(cid:19) (6.17) ≤ e −√ n . (6.18)We note that (6.14) is a union bound, (6.15) follows from the definition of q , (6.16) is the mul-tiplicative form of Chernoff bound (note that the events v ∈ b N V i ( R )( v ∈ V \ R ) are mutuallyindependent), (6.17) follows from (6.3), and (6.18) holds for large enough n by (6.2) and the factthat ε < / and k < n / .All that is left to prove is that asymptotically almost surely G satisfies property 2 in Defini-tion 5.3, that is that every ( r, q ) -neighbour-dense set W ⊆ V is ( tr, r, q ′ , s ) -mostly neighbour-dense.For shortness let P be the event that G satisfies this property. We wish to show that Pr[ ¬ P ] ≤ e − Ω( n ) ,and it turns out that due to our choice of parameters we can afford to use the crude union boundover all n choices of W .To be more specific, let Q ( W ) denote the event that W is ( r, q ) -neighbour-dense. Given an ( r, q ) -neighbour-dense set W ⊆ V we will define a set S W which will be a “candidate witness” ofthe fact that W is ( tr, r, q ′ , s ) -mostly neighbour-dense. First observe that, since W is ( r, q ) -neigh-bour-dense and q ′ ≤ q by (6.5), any set R ⊆ V with | R | ≤ tr and (cid:12)(cid:12) b N W ( R ) (cid:12)(cid:12) ≤ q ′ must be suchthat | R | > r . We will use a sequence of such sets R and construct S W in a greedy fashion. To thisend, the following definition will be useful. A tuple of sets ( R , . . . , R m ) is said to be r -disjoint if (cid:12)(cid:12) R i ∩ (cid:0) S j r . This implies that if | S W | ≤ s then S W witnesses the fact that W is ( tr, r, q ′ , s ) -mostly neighbour-dense. Therefore we have that Pr[ ¬ P ] ≤ Pr[ ∃ W ⊆ V, Q ( W ) ∧ | S W | > s ] . (6.19)Moreover, let W be the collection of all pairs ( W, ~R ) such that W ⊆ V , ~R = ( R , . . . , R ℓ ) for ℓ = ⌈ s/tr ⌉ , R j ⊆ V and < | R j | ≤ tr for each j ∈ [ ℓ ] , and ~R is r -disjoint. Notice that if there20xists an ( r, q ) -neighbour-dense W such that ~R W = ( R , . . . , R m ) and | S W | > s , then m ≥ ℓ and ( W, ( R , . . . , R ℓ )) ∈ W . Furthermore, by definition of ~R W , for every j ∈ [ ℓ ] it holds that (cid:12)(cid:12) b N W ( R j ) (cid:12)(cid:12) ≤ q ′ . Hence we can conclude that Pr[ ¬ P ] ≤ Pr (cid:2) ∃ ( W, ~R ) ∈ W , Q ( W ) ∧ ∀ j ∈ [ ℓ ] , (cid:12)(cid:12) b N W ( R j ) (cid:12)(cid:12) ≤ q ′ (cid:3) (6.20) ≤ n n trℓ max ( W, ~R ) ∈W Pr (cid:2) Q ( W ) ∧ ∀ j ∈ [ ℓ ] , (cid:12)(cid:12) b N W ( R j ) (cid:12)(cid:12) ≤ q ′ (cid:3) (6.21) ≤ n n s max ( W, ~R ) ∈W Pr (cid:2) Q ( W ) ∧ ∀ j ∈ [ ℓ ] , (cid:12)(cid:12) b N W ( R j ) (cid:12)(cid:12) ≤ q n − tδr (cid:3) , (6.22)where (6.22) follows for n large enough from the bound in (6.5).Now fix ( W, ~R ) ∈ W and let R dj (resp. R cj ) be the subset of R j disjoint from (resp. con-tained in) S j ′
10, ST10, SRJ11, SMRH13, SLB14, SLB +
16, TYH + ¨Osterg˚ard’s algorithm ¨Osterg˚ard’s algorithm [ ¨Ost02] is a branch and bound algorithm that usesRussian doll search as a pruning strategy: it considers smaller subinstances recursively and solvesthem in ascending order using previous solutions as upper bounds. This algorithm, which is themain component of the Cliquer software, is often used in practice and has been available onlinesince 2003 [N ¨O03]. Cliquer is also the software of choice to compute maximum cliques in theopen source mathematical software SageMath [S + Cliquer ( G ) algorithm described in Figure 2 is essentially the same as Algorithm 2 in [ ¨Ost02].The algorithm first permutes the vertices of G according to some criteria. Let v , . . . , v n be the enu-meration of V ( G ) induced by said permutation, and V i = { v i , . . . , v n } for i ∈ [ n ] . In practicethis permutation has a large impact on the running time of the algorithm, but for our analysis theknowledge of the specific order is irrelevant.In the main loop (lines 5–8) subgraphs of G are considered and at each iteration the size ofa maximum clique containing only vertices of V i is stored in bounds [ i ] . The algorithm keeps thebest solution (largest clique) found so far in the global variable incumbent which is initially empty.The array bounds and the flag found are global variables. The current growing clique is stored in solution and passed as an argument of the subroutine expand together with the current subgraph H ⊆ G being considered. 22 lgorithm 2 Cliquer ( G ) algorithm Cliquer ( G ) : begin G ← permute ( G ) incumbent ← ∅ for i = n down to do found ← false expand ( G [ V i ∩ N ( v i )] , { v i } ) bounds [ i ] ← | incumbent | return incumbent expand ( H, solution ) : begin while V ( H ) = ∅ do if | solution | + | V ( H ) | ≤ | incumbent | then return i ← min { j | v j ∈ V ( H ) } if | solution | + bounds [ i ] ≤ | incumbent | then return solution ′ ← solution ∪ { v i } V ′ ← V ( H ) ∩ N ( v i ) expand ( H [ V ′ ] , solution ′ ) if found = true then return H ← H \ { v i } if | solution ′ | > | incumbent | then incumbent ← solution ′ found ← true return The main subroutine expand recursively goes through all vertices of H from smallest to largestindex. First note that if the size of the current growing clique plus | H | is not larger than the currentmaximum clique (line 13) then this branch can be cut. Moreover, if v i is the smallest-index vertexin H then V ( H ) ⊆ V i and bounds [ i ] is an upper bound on the size of a maximum clique in H .This implies that this branch can be cut if the size of the current growing clique plus bounds [ i ] isnot larger than the current maximum clique (line 15). If it is larger, the algorithm branches on thevertex v i .First v i is taken to be part of the solution: it is added to (a copy of) the current growing solu-tion, (a copy of) the graph is updated to contain only neighbours of v i and a recursive call is made(lines 16–18). If the recursive call finds a clique larger than the current largest clique, it sets the flag found to true. This allows the algorithm can return to the main routine (line 8) since a maximumclique containing only vertices of V i can be at most one unit larger than a maximum clique contain-ing only vertices of V i +1 . If no larger clique was found, the algorithm then proceeds to the opposite23ranch choice, that is, taking vertex v i to not be in the solution (line 20) and considering the nextvertex in the ordering. If V ( H ) is empty and a larger clique has been found, the best solution so faris updated and the flag found is set to true (lines 22–23).We now argue that the running time of the Cliquer ( G ) algorithm is bounded from below bythe size of a regular resolution refutation of Clique block ( G, k ) up to a constant factor. First note thata straightforward modification of the Cliquer ( G ) algorithm gives an algorithm that determineswhether G contains a block-respecting k -clique.Given a graph G that does not contain a block-respecting k -clique, the last call of the subroutine expand in the main loop (lines 5–8, when i is set to ) can be represented by an ordered decisiontree with labelled leafs. A decision tree is said to be ordered if there exists a linear ordering of thevariables such that if x is queried before y then x ≺ y . In our setting, the order is determined bythe permutation of the vertices, and without loss of generality we assume v i ≺ v j if i < j . Foreach leaf, if R is the set of vertices identified as clique members by the branch leading to this leaf,then the leaf is labelled either by a pair ( u, v ) such that u, v ∈ R and there is no edge between u and v or by an index ℓ ∈ [ k ] such that all vertices in the ℓ th block are outside the clique, or by avertex v i such that i = min { j | v j ∈ N ( R ) } and the largest clique containing only vertices of V i has size at most k − | R | − . For each vertex v i that labels some leaf, we construct the decision treecorresponding to the i th call of the subroutine expand .In order to weave these decision trees into a read-once branching program, at each leaf labelled v i we query all non yet queried vertices v j such that j < i and v j is in the same block as v i . Let B i denote the set of vertices. Observe that taking any vertex in B i to be in the clique yields animmediate contradiction since B i ∩ N ( R ) = ∅ by definition of i . Moreover note that the branchleading to the leaf where all of B i is taken to be outside the clique does not contain any query tovertices in V i . We can therefore identify this leaf with the root of the decision tree corresponding to v i and still maintain regularity. After repeating this procedure at every leaf labelled by some vertex,only leafs labelled by indices ℓ ∈ [ k ] and by pairs ( u, v ) remain, which have a direct correspondenceto falsified clauses of Clique block ( G, k ) . Therefore, the directed graph obtained by this processcorresponds to a read-once branching program that solves the falsified clause search problem on Clique block ( G, k ) and the bound on the running time follows immediately. Colour-based branch and bound algorithms
We consider a class of algorithms which are ar-guably the most successful in practice. An extended survey together with a computational analysisof algorithms published until 2012 can be found in [Pro12] and an overview of algorithms reportedsince then in [McC17]. These algorithms are branch and bound algorithms that use colouring as abounding—and often also as a branching—strategy. The basic idea is that if a graph can be colouredwith ℓ colours then it does not contain a clique larger than ℓ .The MaxClique ( G ) algorithm described in Figure 3, a generalized version of Algorithm 2.1in [McC17], is a basic maximum clique algorithm which uses a colour-based branch and boundstrategy. The algorithm keeps the best solution (largest clique) found so far in the global variable incumbent which is initially empty. The current clique is stored in solution and passed as anargument of the subroutine expand together with the current subgraph H ⊆ G being considered.The subroutine colourOrder ( H ) (line 8) returns an ordering of the vertices in H , say v , v , . . . , v n ,24 lgorithm 3 MaxClique ( G ) algorithm MaxClique ( G ) : begin global incumbent ← ∅ expand ( G, ∅ ) return incumbent expand ( H, solution ) : begin ( order , bounds ) ← colourOrder ( H ) while V ( H ) = ∅ do i ← | V ( H ) | if | solution | + bounds [ i ] ≤ | incumbent | then return v ← order [ i ] solution ′ ← solution ∪ { v } V ′ ← V ( H ) ∩ N ( v ) expand ( H [ V ′ ] , solution ′ ) H ← H \ { v } if | solution ′ | > | incumbent | then incumbent ← solution ′ return and for every i ∈ [ n ] an upper bound on the number of colours needed to colour the graph inducedby vertices v to v i .The vertices are then considered in reverse order. If the vertex v is being considered and the sizeof the current growing clique plus the (upper bound on the) number of colours needed to colour theremaining graph is not larger than the current maximum clique (line 11) then this branch can be cut.If it is larger, the algorithm branches on the vertex v . First v is taken to be part of the solution: v isadded to (a copy of) the current growing solution, (a copy of) the graph is updated to contain onlyneighbours of v and a recursive call is made (lines 13–15). If the recursive call finds a clique largerthan the current largest clique, the best solution so far is updated (line 17). The algorithm proceedsto the opposite branch choice, that is, considering vertex v not in the solution (line 16). Returningto the loop the algorithm continues to consider the next vertex in the ordering.It was reported in [CZ12] that it is possible to capture the algorithms for solving the maximumclique problem in [CP90, Fah02, TS03, TK07, KJ07, TSH +
10] in a same framework. The gen-eral algorithm they present is an iterative version of the
MaxClique ( G ) algorithm. We observethat MaxClique ( G ) captures also the more recent algorithms in [ST10, SRJ11, SMRH13, SLB14,SLB +
16, TYH + k -clique formula, we assume that thecolouring algorithm and the graph operations take constant time and prove the lower bound for this25eneral framework. Moreover, we can assume that optimal colouring bounds and optimal orderingof vertices are given.We now argue that the running time of the MaxClique ( G ) algorithm is bounded from below bythe size of a regular resolution refutation of Clique block ( G, k ) up to a multiplicative factor of k n O(1) .We first note that a straightforward modification of the
MaxClique ( G ) algorithm gives an algorithm,which we refer to as Clique ( G, k ) , that determines whether G contains a k -clique. Given a graph G that does not contain a k -clique, an execution of Clique ( G, k ) can be represented by a searchtree with leafs labelled by a subgraph H ⊆ G of potential clique-members and a number q such thatthe branch leading to this leaf has identified k − q clique members, has not queried any vertex of H ,and H is ( q − -colourable. Note that a read-once branching program can simulate this search treeand, by Proposition 3.3 and the equivalence between read-once branching programs and regularresolution, at each leaf establish that H does not contain a q -clique in size at most q · q · | V ( H ) | .The bound on the running time follows directly.Observe that establishing that H does not contain a q -clique is done in a read-once fashion byquerying only vertices of H . Since the vertices of H were not queried earlier on this branch, thewhole branching program is read-once. In this paper we prove optimal average-case lower bounds for regular resolution proofs certifying k -clique-freeness of Erd˝os-R´enyi graphs not containing k -cliques. These lower bounds are alsostrong enough to apply for several state-of-the-art clique algorithms used in practice.The most immediate and compelling question arising from this work is whether the lowerbounds for regular resolution can be strengthened to hold also for general resolution. A closerstudy of our proof reveals that there are several steps that rely on regularity. However, there is noconnection per se between regular resolution and the abstract combinatorial property of graphs thatwe show to be sufficient to imply regular resolution lower bounds. Thus, it is tempting to speculatethat this property, or perhaps some modification of it, might be sufficient to obtain lower boundsalso for general resolution. If so, a natural next step would be to try to extend the lower boundfurther to the polynomial calculus proof system capturing Gr¨obner basis calculations. It is worthmentioning that proving a general resolution lower bound of n Ω( k ) for the k -clique formula wouldhave interesting consequences in parameterized proof complexity [DMS11].Another intriguing question is whether the lower bounds we obtain asymptotically almost surelyfor random graphs can also be shown to hold deterministically under the weaker assumption thatthe graph has certain pseudorandom properties. Specifically, is it possible to get an n Ω(log n ) lengthlower bound for the class of Ramsey graphs? A graph on n vertices is called Ramsey if it has no set of ⌈ n ⌉ vertices forming a clique or an independent set. It is known that for sufficiently large n arandom graph sampled from G ( n, / is Ramsey with high probability. Is it true that for a Ramseygraph G on n vertices the formula Clique ( G, ⌈ n ⌉ ) requires (regular) resolution refutations oflength n Ω(log n ) ? The main difficulty towards adapting our argument to this setting is that Ramseygraphs are, in some sense, less well structured than random graphs. For example, a random graphplus a constant number of isolated vertices is, with high probability, still a Ramsey graph, but it26o longer satisfies the first property of clique-denseness (Definition 5.3). This particular problemcan be circumvented using a result from [PR99, Theorem 1]—as was done in [LPRT17] to obtain alower bound for tree-like resolution—but proving that a Ramsey graph satisfies the second propertyof clique-denseness, or some suitable version of it, seems significantly more challenging. Acknowledgements
This work has been a long journey, and different subsets of the authors wantto acknowledge fruitful and enlightening discussions with different subsets from the following listof colleagues: Christoph Berkholz, Olaf Beyersdorff, Nicola Galesi, Ciaran McCreesh, Toni Pitassi,Pavel Pudl´ak, Ben Rossman, Navid Talebanfard, and Neil Thapen. A special thanks to Shuo Pangfor having pointed out an inaccuracy in the probabilistic argument in Section 6 and having suggesteda fix.The first, second, and fourth authors were supported by the European Research Council underthe European Union’s Horizon 2020 Research and Innovation Programme / ERC grant agreementno. 648276 AUTAR.The third and fifth authors were supported by the European Research Council under the Euro-pean Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement no. 279611as well as by Swedish Research Council grants 621-2012-5645 and 2016-00782, and the secondauthor did part of this work while at KTH Royal Institute of Technology supported by the samegrants. The last author was supported by the Russian Foundation for Basic Research.
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