Strongly refuting all semi-random Boolean CSPs
aa r X i v : . [ c s . CC ] S e p Strongly refuting all semi-random Boolean CSPs
Jackson Abascal ∗ Venkatesan Guruswami † Pravesh K. Kothari ‡ Computer Science DepartmentCarnegie Mellon UniversityPittsburgh, PA 15213
Abstract
We give an efficient algorithm to strongly refute semi-random instances of all Boolean con-straint satisfaction problems. The number of constraints required by our algorithm matches(up to polylogarithmic factors) the best known bounds for efficient refutation of fully randominstances. Our main technical contribution is an algorithm to strongly refute semi-random in-stances of the Boolean k -XOR problem on n variables that have e O ( n k/ ) constraints. (In asemi-random k -XOR instance, the equations can be arbitrary and only the right hand sides arerandom.)One of our key insights is to identify a simple combinatorial property of random XOR in-stances that makes spectral refutation work. Our approach involves taking an instance that doesnot satisfy this property (i.e., is not pseudorandom) and reducing it to a partitioned collectionof 2-XOR instances. We analyze these subinstances using a carefully chosen quadratic form asproxy, which in turn is bounded via a combination of spectral methods and semidefinite pro-gramming. The analysis of our spectral bounds relies only on an off-the-shelf matrix Bernsteininequality. Even for the purely random case, this leads to a shorter proof compared to the onesin the literature that rely on problem-specific trace-moment computations. ∗ Research supported in part by NSF grant CCF-1563742. [email protected] † Research supported in part by NSF grant CCF-1908125. [email protected] ‡ [email protected] ontents k -XOR for k > d -bounded and 2-XOR cases for general partitioned 2-XOR refutation . . 145.3 Semi-random 2-XOR Refutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 The d -Bounded Case: Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . 17 References 23
Introduction
The study of random constraint satisfaction problems (CSPs) is a major thrust in the theoryof average-case algorithm design and complexity. Investigations into this problem are centralto disparate areas including cryptography [ABW10], proof complexity [BSB02], hardness of ap-proximation [Fei02a], learning theory [DLS14], SAT-solving [BK], statistical physics [CLP02], andcomplexity theory [BKS13].For a predicate P : { , } k → { , } , an instance of CSP( P ) consists of a set of Boolean variablesand a collection of constraints each of which applies P to a tuple of k literals (variables or theirnegations). The goal is to understand the maximum fraction of constraints that can be satisfiedby any Boolean assignment to the variables. In the fully random model, the k -tuples and theliteral patterns are chosen independently and uniformly at random. In the semi-random model,the k -tuples are arbitrary (i.e., “worst-case”) and only the negation patterns are chosen randomly.When the number m of constraints is much larger than the number n of variables, such instancesare unsatisfiable with high probability. The algorithmic goal is to find an efficient algorithm that,for most instances, finds a refutation —an efficiently verifiable certificate that no assignment cansatisfy all constraints ( weak refutation). More stringently, we can ask for a strong refutation that,for some absolute constant δ >
0, the instance is not “(1 − δ )-satisfiable” in that no assignmentcan satisfy even a fraction (1 − δ ) fraction of constraints.The question of refuting fully-random CSPs was first considered in the seminal work ofFeige [Fei02a]. Recently, this effort has had a series of exciting developments giving both new effi-cient algorithms [AOW15, RRS16, BGL17] and nearly matching lower-bounds [BCK15, KMOW16,FPV15] in several strong models of computation.The focus of this work is the more challenging semi-random CSP model. The semi-random(and the related, “smoothed”) CSP model was formalized by Feige [Fei07] following a long line ofwork [BS95, FK01] on studying semi-random instances of natural problems. In that work, Feigegave a weak refutation algorithm for semi-random 3-SAT and noted:
Our motivation for studyingsemi-random models is that they are more challenging than random models, lead to more robustalgorithms, and lead to a better understanding of the borderline between difficult and easy instances.
For an introduction and recent survey of semi-random models, see the chapter [Fei19] in “BeyondWorst-case Analysis of Algorithms.”The question of semi-random refutation was revived recently in the work of Allen, O’Donnell andWitmer [AOW15] who gave state-of-the-art efficient strong refutation algorithms for fully randomCSPs and explicitly asked whether their algorithm could extend to the more challenging semi-random setting. This direction was further fleshed out in Witmer’s PhD thesis [Wit17] (see Chapter5). Two years later, a sequence of exciting works [LT17, Lin16, LV16, Lin17, AS16, GJLS20] incryptography related this question to the security of arbitrary instantiations of Goldreich’s localpseudorandom generator [Gol00] and showed that it implies indistinguishability obfuscation understandard assumptions. In their work on refuting hardness assumptions in this context, [BBKK18]note that better algorithms for semi-random refutation imply stronger attacks on the candidatelocal pseudo-random generators in cryptography.
In this work, we resolve the question of semi-random CSP refutation by giving polynomial timealgorithms that match the guarantees on fully random CSP instances.1 heorem 1.1.
Fix a predicate P : { , } k → { , } . Let r ( P ) = E x [ P ( x )] over a uniformly random x ∈ { , } k , let d ( P ) be the degree of P as a multilinear polynomial, and let i ( P ) be the largestinteger t for which P − (1) supports a t -wise uniform distribution, i.e., a distribution whose jointmarginals on every t variables is uniform.There is a polynomial time algorithm that given an n -variable semi-random instance of CSP( P ) with m > n ( i ( P )+1) / poly(log n ) constraints, with high probability certifies that it is not (1 − δ ) -satisfiable for some δ = Ω k (1) .Further, when the number of constraints exceeds n d ( P ) / poly((log n ) /ε ) , the algorithm certifiesthat the instance is not ( r ( P ) + ε ) -satisfiable (the “right” bound). For the significance of the quantity i ( P ) in the context of CSP refutation, see [AOW15] whoshowed strong refutation of random CSPs for a similar number of constraints. Our work thusextends the state-of-the-art in CSP refutation from random to semi-random CSPs.
It’s all about XOR.
The question of refuting arbitrary Boolean CSPs immediately reduces torefuting the specific k -XOR CSPs for all k via Feige’s [Fei02a] “XOR principle” formalized in thework of [COCF10] (see also, Theorem 4.9 in [AOW15]). This involves writing the Boolean pred-icate as a multilinear polynomial (its discrete Fourier transform) and refuting each CSP instancecorresponding to a term in this expansion. For even k , Witmer and Feige [Wit17] observed thatan idea from [Fei07] already gives a strong semi-random refutation algorithm (via “discrepancy”)that matches the best-known guarantees in the fully-random model. For odd k , however, the bestknown efficient algorithm succeeds only when the number of constraints is ˜Ω( n ⌈ k/ ⌉ ) which is largerby a factor of √ n when compared to the fully random case. Our main technical contribution is toresolve this deficiency for odd k : Theorem 1.2 (Main, detailed version appears as Theorem 4.2) . There is a polynomial time algo-rithm that given an n -variable semi-random k -XOR instance for odd k with m > n k/ poly((log n ) /ε ) equations, with high probability certifies that it is not (1 / ε ) -satisfiable. A priori, the issue of odd vs even arity XOR might appear like a mere technical annoyance.However, even for the fully random case, strongly refuting odd-arity XOR turns out to be trickier and was resolved only in the recent work of Barak and Moitra [BM15] by means of an intense argu-ment based on the trace-moment method. In his work on weak refutation of semi-random 3-SAT,Feige manages to skirt this issue by observing that proving a slightly non-trivial (but still 1 − o n (1))upper-bound on the value of a semi-random 3-XOR instance is enough. He accomplishes this goalby a sophisticated combinatorial argument (that improved on a work of Naor and Verstrte [NV08])based on the existence of small even covers in sufficiently large hypergraphs.To obtain the strong refutation guaranteed by our work, we replace this combinatorial argumentby a careful combination of spectral bounds and semidefinite programming. As explained in moredetail in Section 2, we identify a certain pseudorandomness property (which holds for random k -XOR instances) under which strong refutation follows from spectral bounds of certain matricesthat are sums of independent random matrices. To tackle the semi-random case, we decompose theinstance into various structured sub-instances and associated quadratic forms whose sum boundsthe optimum of the CSP instance. While the spectral norm of the matrix associated with a sub-instance might be large, for the CSP refutation we only care about vectors with commensurately In k -XOR CSPs, the predicate P computes the XOR of the k input bits. see discussion following Theorem 2.1 on Page 4 of [AOW15]. ℓ -norm. We also need to tackle variables which participate in too many constraintsseparately, and we do this by forking off an auxiliary 2-XOR instance which is refuted using theGrothendieck inequality.Our work leads to many natural open questions. An important one is whether we can remove thepolylogarithmic multiplicative factors in Theorem 1.2 and strongly refute with O ( n k/ ) constraints.We note that for the case of odd k , this is open even in the case of random k -XOR. While thismight seem like a rather technical question, it is interesting because a resolution would likely callfor more sophisticated random matrix theory methods. In this regard, the work of [DMO +
19] onrandom NAE-3SAT might be worth highlighting.
Here, we give a brief survey of the long line of work on average-case complexity of CSPs.There is a long line of work on studying algorithms and lower-bounds for refuting/solvingrandom CSPs and connections to other computational problems. We only include the highlightshere and point the reader to [KMOW16] for a more extensive survey. Feige [Fei02a] made a fruitfulconnection between average-case complexity of random CSPs and hardness of approximation andformulated his “R3SAT” hypothesis that conjectures that there’s no polynomial time algorithm forstrong refutation of 3-SAT with m = Cn constraints for some C ≫ .
27. This and its generalizationto other CSPs have served as starting points of reductions that have yielded hardness results inseveral domains including cryptography, learning theory, and game theory [Fei02b, Ale03, BKP04,DFHS06, Bri08, AGT12, DLSS13, DLS14, Dan15, ABW10, App13]. Very recently, variants ofthe assumptions were used to give candidate constructions of indistinguishability obfuscation incryptography [LT17].Coja-Oghlan [COGL07] gave the first δ (i.e. certifying a bound of 1 − δ for some fixed con-stant δ >
0) refutation algorithms for 3 and 4-SAT. Early works used combinatorial certificatesof unsatisfiability for random 3SAT instances [Fu96, BKPS02]. Goerdt and Krivilevich [GK01]were the first to examine spectral refutations with quick follow-ups [FGK05, GL03]. More recently,Allen, O’Donnell, Witmer [AOW15] gave a polynomial time algorithm for refuting random CSPsand Raghavendra, Rao and Schramm [RRS16] extended it to give sub-exponential algorithms withnon-trivial guarantees. Both these algorithms are captured in the sum-of-squares semidefinite pro-gramming hierarchy.There’s a long line of work proving lower-bounds for refuting random CSPs in proof complexitybeginning with the work of Chv´atal and Szemer´edi [CS88] on resolution refutations of random k -SAT. Ben-Sasson and Wigderson [BSW01, BS01] and Ben-Sasson and Impagliazzo and Alekhnovichand Razborov [BSI99, AR01] further extended these results.One of the first lower bounds for random CSPs using SDP hierarchies was given by Buresh-Oppenheim et al. [BOGH + + (LS + ) proof system cannotrefute random instances of k -SAT with k > m/n . Alekhnovich,Arora, and Tourlakis [AAT05] extended this result to random instances of 3-SAT. The strongestresults along these lines involve the Sum of Squares (AKA Positivstellensatz or Lasserre) proofsystem (see e.g., [OZ13, BS14, Lau09, FKP19] for surveys concerning SOS). Starting with Grig-oriev [Gri01], a number of works [Sch08, Tul09, BGMT12, OW14, TW13, MW16] have studied thecomplexity of SoS and the related Sherali-Adams and Lovasz-Schrijver proof systems for refutingrandom instances of CSPs. The strongest known results in this direction are due to the recent3orks [BCK15] and [KMOW16] who gave an optimal 3-way trade-off between refutation strength,number of constraints and a certain complexity measure of the underlying predicate.Beyond semialgebraic proof systems and hierarchies, Feldman, Perkins, and Vempala [FPV15]proved lower bounds for refutation of CSP( P ± ) using statistical algorithms when P supports a( t − In this section, we give a high-level overview of the main ideas in our refutation algorithm byfocusing on the case of k -XOR. This immediately implies refutation algorithms for all BooleanCSPs using Feige’s standard “XOR principle” [Fei02a].We will use the ± k -XOR instance φ as composed of a k -uniform hypergraph with m edges specifying the tuples that are constraint constrained, a set of RHS’s r ∈ {± } m . We allowrepeated edges in the hypergraph; this is necessary for the reduction from general semi-randomCSP to XOR.For an assignment x ∈ {± } m , definebias φ ( x ) = 1 m m X j =1 r j · Y i ∈ C j x i , and let bias φ := max x ∈{± } n bias φ ( x ) . Then, the fraction of constraints val φ ( x ) satisfied by an assignment x equalsval φ ( x ) = 1 + bias φ ( x )2 . A semi-random refutation algorithm takes H, r and outputs a value v such that1. v > bias φ , and2. P r ∈{± } m [ v ε ] > . . Observe that one can obtain a weak refutation algorithm by simply running Gaussian elimination on the k -XORsystem and outputting 1 − /m if the system is insoluble. However such algorithms do not yield even weak refutationalgorithms for other CSPs and are not of interest to us in this work. m ≫ n/ε , there is an inefficient algorithm that succeeds in both these goals: thealgorithm simply outputs bias φ . To obtain efficient refutation algorithms, we need to come upwith polynomial-time computable upper bounds on bias φ that are good approximations to bias φ for semi-random φ .All algorithms for refuting CSPs (including ours) are captured by a canonical semidefiniteprogram (an appropriate number of levels of the sum-of-squares hierarchy). However, our analysiscan be viewed as giving a stand-alone efficient algorithm that uses only a basic SDP and spectralnorm bounds on appropriately (and efficiently) generated matrices. We will adopt this perspectivein this overview. Our algorithms apply to settings where the constraint hypergraph H is k -uniformbut can have repeated hyperdges (i.e. is a k -uniform multi-hypergraph). This level of generality isrequired in order to obtain our semi-random refutation algorithm for all CSPs. However, for thepurpose of the overview, we will restrict ourselves to simple k -uniform hypergraphs H . -XOR Instances For k = 2, there are two natural strategies that yield optimal (up to absolute constants) refutationalgorithms. We explain them in detail because one can view all known strong refutation algorithmsas simply reducing to 2-XOR and then using one of these two strategies.The key observation is that bias φ ( x ) is a quadratic form: bias φ ( x ) = x ⊤ Ax where A ( i, ℓ ) = A ( ℓ, i ) = r j / C j = { i, ℓ } for some j and 0 otherwise. This immediately allows two natural,efficient upper-bounds for bias φ ( x ).1. Spectral Method.
We observe that bias φ ( x ) = x ⊤ Ax k x k k A k . Thus, bounding thespectral norm of A , which is efficiently computable, immediately gives an upper-bound onbias φ . When φ is random, A is a sparse random matrix and known results [BvH16] implythat k A k O ( √ log n ) with high probability. This immediately yields a refutation algorithmthat succeeds whenever m ≫ n log n/ε .2. Semidefinite programming.
This is a refinement of the above idea and relies on thefollowing reasoning: bias φ = max x ∈{± } n x ⊤ Ax max x,y ∈{± } n x ⊤ Ay . This latter quantityis known as the “infinity to 1” norm of A and the well-known Grothendieck inequality showsthat the value of a natural semidefinite programming relaxation (see Lemma 3.3) sdp( φ )approximates it within a factor <
2. It is easy to argue using the Chernoff bound followedby a union bound that max x,y ∈{± } n x ⊤ Ay O ( p n/m ). Thus, bias φ sdp( φ ) O ( p n/m )with high probability giving a refutation algorithm that works whenever m ≫ n/ε . k -XOR for k > When k >
2, a natural strategy is to simply find a “higher-order” (tensor) analog of theGrothendieck inequality/eigenvalue-bound we used above. However, there are no tractable analogsof these bounds that provide a O (1) approximation. Indeed, the best known algorithms [KN08]lose poly( n ) factors in approximation and such a loss might be necessary. As a result, all knownefficient refutation algorithms rely on reducing k -XOR instances to 2-XOR via a linearization trick. The case of even k . For even k , this trick considers a 2 XOR instance with (cid:0) nk/ (cid:1) variables,each corresponding to k/ m ≫ n k/ , in the linearized5pace, we have n k/ variables and ≫ n k/ constraints with independently random RHS. Using asimple argument based on Chernoff + union bound immediately shows that the resulting 2-XORinstance has value at most O ( p n k/ /m ). One can then use either of the above two methods toobtain a good approximation to this value and get a refutation algorithm. In particular, spectralrefutation gives an algorithm that succeeds whenever m ≫ n k/ (log n ) /ε for random k -XORinstances. This method, however, cannot yield any non-trivial refutation algorithm in the semi-random case. Fortunately, the SDP method continues to work fine giving a semi-random refutationalgorithm for even k that succeeds whenever m ≫ n k/ /ε . The case of odd k and the main challenge. For odd k , the situation is more complicatedsince there is no natural “balanced” linearization. The natural “unbalanced” linearization yieldsa 2-XOR instance on n ( k +1) / variables yielding a refutation algorithm that succeeds whenever m ≫ n ( k +1) / /ε that is off from our goal by a √ n factor. For the case when H is random ,Barak and Moitra [BM15] found a clever way forward using their Cauchy-Schwarz trick . This ideaconstructs a 2( k − φ ′ with n k − variables from the input k -XOR instance φ , relatesthe value of φ ′ to φ and then refutes φ ′ (which is an even arity XOR instance) via linearizationfollowed by the spectral method above. Specifically, φ ′ takes every pair of constraints in φ thatshare a variable and “XORs” them to derive a new, 2( k −
1) XOR constraint. For k = 3, thisresults in a 4-XOR instance.Notice that a priori, it is not clear that the 2-XOR instance produced by linearizing the 2( k − φ ′ produced via this route has a small true value! Indeed, the resulting constraintshave dependent right hand sides with only m ≈ n k/ bits of randomness while the number ofvariables in the linearized space is n k − . Thus, we cannot execute a Chernoff + union bound styleargument to upper-bound the true value as we did for the even k case above. Nevertheless, theproof of Barak and Moitra cleverly exploits the randomness in the constraint graph H to prove this(via the spectral upper-bound) using a somewhat intense calculation using the trace method.Alas, this strategy is a non-starter in our case. The spectral bound that the technique of Barakand Moitra proves is simply not even true in the semi-random case, in general. Note that the truevalue of this 2-XOR instance may still be small—it is just that the spectral upper bound is tooloose to capture it. Indeed, no known method (including ours) gives an upper bound on the valueof this particular 2-XOR instance in the semi-random case. Aside: Feige’s semi-random refutation via combinatorial method.
It is instructive to notethat Feige did manage to obtain an efficient weak refutation algorithm for semi-random strongly refuting 3-XOR and showing instead that for weakrefutation of 3-SAT, one only needs to weakly refute semi-random 3-XOR with a value upper boundof 1 − Ω(1) / (cid:16) m − ˜ O ( n / ) (cid:17) . This latter bound can be obtain by a combinatorial method that, ata high-level, involves breaking the m constraints into groups of size ≈ n / with an appropriatecombinatorial structure (called even covers ) each and proving that each of them is individuallyunsatisfiable.Indeed, up until this work, the case of semi-random refutation of odd arity XOR, while occurringin several applications has remained unresolved. Feige’s argument applies to a slightly more general “smoothed” setting. We have not (yet) investigated if ourapproach also extends to this setting.
6e now describe the key insights that lead to our method. While our overall algorithm couldbe thought of as simply analyzing the value of a natural SDP (see Section 5), it is helpful to breakopen the analysis and view the algorithm as decomposing the instance and then using spectralmethod and SDP discussed above for appropriately chosen pieces. We adopt this perspective inthis overview. We will focus on the case of k = 3 to keep things simple here. -XOR refutation In order to motivate our plan, it is helpful to give a quick and painless proof (that does not usethe trace-moment method) of the spectral refutation for refuting random pseudo-random instances and show that refutingan arbitrary , this proof willalso allow us to concretely explain both the decoupling and linearization steps.An instance of 3-XOR with m constraints can be described by a size n collection of n × n matrices as follows. Pick an arbitrary ordering on the n variables. Concretely, define B j,k = B k,j = r ( i, j, k ) / x i x j x k = r ( i, j, k ) appears in φ and i < j and i < k in thechosen ordering - in this case, we say that the constraint was “colored” by color i . Note that everyconstraint is colored by exactly one of the n colors. We can thus write:bias φ ( x ) = 1 m n X i =1 x i · x ⊤ B i x . (2.1) Decoupling.
Next, the idea is to apply Cauchy-Schwarz inequality on the RHS above and use x i = 1 for every i to conclude that:bias φ ( x ) m n X i =1 x i n X i =1 (cid:16) x ⊤ B i x (cid:17) = nm n X i =1 (cid:16) x ⊤ B i x (cid:17) . Observe that one can view this step as “decoupling” the smallest variable (in the ordering chosenabove) from the other two in every constraint.
Linearization.
Next, we observe that ( x ⊤ B i x ) = ( x ⊗ ) ⊤ ( B i ⊗ B i ) x ⊗ where x ⊗ is the n -dimensional vector indexed by ordered pairs of the n variables and x ⊗ ( u, v ) = x u x v for every u, v ∈ [ n ]. This allows us to conclude:max x ∈{± } n bias φ ( x ) nm max y ∈{± } n n X i =1 y ⊤ B i ⊗ B i y nm k y k (cid:13)(cid:13)(cid:13)X i n B i ⊗ B i (cid:13)(cid:13)(cid:13) = n m (cid:13)(cid:13)(cid:13)X i n B i ⊗ B i (cid:13)(cid:13)(cid:13) . (2.2)Observe that we have arrived at a 2 -XOR instance and a spectral upper-bound on its value. Thisis precisely the instance and the spectral upper-bound that the proofs of Barak and Moitra [BM15](and Allen,O’Donnell and Witmer [AOW15]) generate. The spectral upper-bound is analyzed usinga somewhat complicated trace-moment method based analysis in those works. Enter Matrix Bernstein.
Our analysis of the RHS of (2.2) needs two simple observations: We note [WAM19] recently found a similar simple proof for the case of even arity random
XOR.
7. the matrices B i ⊗ B i are independent for each i and E B i ⊗ B i = 0, and2. max i n (cid:13)(cid:13)(cid:13) B i ⊗ B i (cid:13)(cid:13)(cid:13) = k B i k O (1) by observing that a pair of variables can appear in at most O (1) constraints with high probability.We now want to apply the following standard matrix concentration inequality to upper-bound (cid:13)(cid:13)(cid:13)P i n B i ⊗ B i (cid:13)(cid:13)(cid:13) . Fact 2.1 (Matrix Bernstein Inequality, [Tro12] Theorem 1.4) . Let Z , Z , . . . , Z k be mean , sym-metric d × d matrices such that (cid:13)(cid:13)(cid:13) Z i (cid:13)(cid:13)(cid:13) R almost surely. Then, for all t > , P (cid:2) (cid:13)(cid:13)(cid:13)X i k Z i (cid:13)(cid:13)(cid:13) > t (cid:3) d exp( − t / σ + Rt/ , where σ = (cid:13)(cid:13)(cid:13)P i k E Z i (cid:13)(cid:13)(cid:13) is the norm of the matrix variance of the sum. To apply this inequality, we need to estimate the “variance parameter” σ . This is where wedo need to do a simple but very helpful combinatorial observation.Let deg i ( v ) be the number of constraints in φ that are colored by i and contain variable v .Observe that for a random 3-XOR instance with m ≪ n , removing at most o ( m ) constraints(or equivalently, losing an additional o (1) in the certified upper-bound) ensures that every pair ofvariables appears in at most 1 constraint in φ . As a result, deg i ( v ) ∈ { , } for every i and v .Thus, each row of B i can have at most 1 non-zero entry and as a result, the matrix ( B i ⊗ B i )( B i ⊗ B i ) ⊤ = ( B i ⊗ B i ) is a diagonal matrix with ( v, v ′ ) entry given by deg i ( v ) deg i ( v ′ ). Thus,the diagonal entry at say ( v, v ′ ) of the matrix P i ( B i ⊗ B i ) equals P i n deg i ( v ) deg i ( v ′ ) . (cid:13)(cid:13)(cid:13)P i B i ⊗ B i (cid:13)(cid:13)(cid:13) O (log n ).This immediately allows us to certify with probability at least 0 .
99, a bound of O ( n log n/m ) ≪ ε on bias φ whenever m ≫ n / log n/ε . That finishes the proof! Key lesson.
Here’s the key observation in our analysis of the random case above: the matrices B i ⊗ B i are independent even when the constraint graph H is worst-case! Thus, as long as we cancontrol the “variance parameter” σ - we can repeat the analysis from the random case above. We now turn to describing our high level approach to accomplish the main goal in this paper,refuting semi-random instances of k -XOR. Our approach for semi-random refutation is directly inspired from the above argument. First, wemove to a variant of 3-XOR that we call partitioned 2-XOR via the following reasoning: we first Technically, E B i ⊗ B i is non-zero due to the presence of a small (that amount to a lower-order term) of entriesof non-zero expectation. See the proof of Lemma 5.7 for details. C i = ( a, b, c ) ∈ E by one of the three variables in it arbitrarily, say a . Wethen collect all the constraints of color a in a group G a . Note that G a is a graph of pairs of variables.Observe, then, that the following relationship immediately holds:bias φ = max x ∈{± } n m X i m r i x C i max x,y ∈{± } n m X a ∈ [ n ] X C j \ a ∈ G a r j y a x b,c max x ∈{± } n m X a ∈ [ n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X C j \ a ∈ G a r j x b,c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Note that the key difference from the usual 2-XOR problem is that in the instance above, weare interested in maximizing the sum of the absolute values of the subinstance in each group. Suchan instance is an example of the more general partitioned ℓ -norm of the value of the subinstances within each color group viasome Boolean assignment.To proceed, we identify the central feature of the partitioned 2-XOR instance produced froma random 3-XOR instance that led to the success of the above proof strategy: the butterfly degree of a pair of variables (defined below). We use it to define the class of pseudorandom constraintgraphs H as those with bounded butterfly degree. Observe that even though σ depends on boththe constraint graph and the “right hand sides,” the butterfly degree we define below (and used inthe proof for the random case above) is a function only of the constraint graph. Definition 2.2 (Butterfly Degree and Pseudorandom Constraint graphs) . For any v , let deg i ( v )denote the number of constraints in the i -th 2-XOR instance (in the input partitioned 2-XORinstance) that include the variable v . For a pair of variables v, v ′ the butterfly degree of v, v ′ is n P i n deg i ( v ) deg i ( v ′ ). See Figure 1 for a visual depiction of this quantity. A partitioned 2-XORinstance is said to be pseudorandom if the butterfly degree of every pair of variables is at most O (1 /ε ). v ′ u ′ vu i Figure 1: The butterfly degree counts diagrams of this form for fixed v, v ′ , where ( i, v, u ) and( i, v ′ , u ′ ) each specify a unique constraint (e.g. x v x u = y i ).A straightforward generalization of the argument from the previous subsection gives a refutationalgorithm for all pseudorandom instances. -XOR and structured -XOR When a partitioned 2-XOR instance is not pseudorandom , we observe that there must be deg i ( v )that are larger than the average for some i, v . Our main idea here is to partition the input instanceinto two parts: one where all deg i ( v )’s are O (1 /ε ) and the other where some deg i ( v ) is large.For the second part, we linearize to obtain a bipartite n variables (corresponding to y above) and the other side containing n variables. We then showthis bipartite derived instance preserves the original value and thus, via the analysis based on theGrothendieck inequality above, we can use an SDP to efficiently refute this instance.9he first part requires more work and is refuted by a carefully chosen sequence of spectral upper-bounds that utilize the Booleanity of the assignment to prove progressively tighter upper-boundson the effective ℓ norm of the assignment. First, we use a carefully chosen weight for each of the2-XOR subinstances and then apply the Cauchy-Schwarz inequality. Intuitively, this is to ensurethat the Cauchy-Schwarz trick above is applied in the “tightest” possible parameter regime. Theseweights turn out to be almost equal in a partitioned 2-XOR instance generated from a random 3-XOR instance, up to a scaling factor. Thus, our matrix is essentially the same as the one employedabove (and in [BM15]) when specialized to the random 3-XOR setting but is, in general, differentdepending on the constraint graphs H . This weighting does lead to an additional step of arguingthat the value of the weighted instance can be related to the value of the unweighted instance.Next, we consider the matrix whose quadratic form captures the value of the above weighted4-XOR instance produced by the Cauchy-Schwarz trick. We decompose this matrix further intopoly log n pieces so as to populate rows with similar (up to constants) ℓ norm in piece. Ourstrategy is to use a spectral norm upper bound on each of this piece - however, a naive spectralnorm bound does not suffice and is too large. Our key observation is that the upper-bound on thevalue of CSP instance so produced must be smaller than this estimate because the effective ℓ normof the assignment vector must be lower. This is because we can argue an appropriate geometricallydecreasing upper-bound on the row-sparsity of the matrix in each piece of this decomposition. Below we present some notation we will use throughout this paper. • For n ∈ Z > , define [ n ] = { , . . . , n } . • A vector v ∈ R n is Boolean if all of its entries are either 1 or -1. • The spectral norm k M k is the maximum value of x ⊤ M y over all unit vectors x and y , orequivalently the maximum eigenvalue or singular value of M . • The “infinity to 1” norm k M k ∞→ is the maximum value of x ⊤ M y over all Boolean vectors x and y . • A multiset S is a set which allows repeated elements. We think of identical multiset elementsas ordered and distinguishable, so a function f : S → X may take different values on differentcopies of an element in S . Similarly, when performing an iterative operation over S (forexample, P s ∈ S f ( s )) we iterate over each copy of an element individually. If S, T are multisetsthen S + T is the disjoint union of their contents, so in particular | S + T | = | S | + | T | .A k -XOR instance φ = ( V, E, r ) is specified by a variable set V , a constraint multiset E ofelements in (cid:0) Vk (cid:1) , and a sign function r : E → {± } . We call the instance semi-random if the edgemultiset can be arbitrarily chosen but r is chosen uniformly at random from all functions E → {± } . An assignment to ψ is a vectors x ∈ {± } V . We say that a constraint e ∈ E is satisfied when x e r ( e ) = 1. Define val φ ( x ) as the maximum fraction of satisfied constraints with assignment x , anddefine val φ = max x val φ ( x ).A partitioned 2-XOR instance ψ = ( V, ℓ, E, r ) is specified by a variable set V , an integer ℓ denoting the number of parts, a constraint multiset E of elements in [ ℓ ] × (cid:0) T (cid:1) , and a sign function r : E → {± } . We often write r i ( e ) = r ( i, e ). We call the instance semi-random if E is arbitr ary10nd r is chosen uniformly at random from all functions E → {± } . An assignment to ψ is twovectors x ∈ {± } V and y ∈ {± } ℓ . We say that a constraint ( i, e ) ∈ E is satisfied by ( x, y ) when x e y i r i ( e ) = 1. The set E encodes a partition of a multiset of edges in (cid:0) V (cid:1) , where the i -th part maybe realized explicitly as T i = { e ∈ (cid:0) V (cid:1) | ( i, e ) ∈ E } . Define val ψ ( x, y ) as the maximum fraction ofsatisfied constraints with assignment ( x, y ), and defineval ψ = max x,y val ψ ( x, y ) . We now present some known results which will our proofs will rely on. The first bound is a loose,but useful fact.
Lemma 3.1.
Let X ∈ R a × b . Then the spectral norm of X satisfies k X k max max i a b X j =1 | X i,j | , max j b a X i =1 | X i,j | . Equivalently, k X k is upper bounded by the maximum ℓ norm of any row or column of X .Proof. This is a corollary of the Gershgorin circle theorem applied to the block matrix (cid:2) XX ⊤ (cid:3) ,since the spectral norm of this block matrix equals the the spectral norm of X .The most important inequality underlying our work is the matrix Bernstein inequality. Thematrix Bernstein inequality is useful for bounding the spectral norm of random matrices which canbe easily decomposed as a sum of matrices with uniformly bounded spectral norms. Theorem 3.2 (Rectangular Matrix Bernstein, [Tro12] Theorem 1.6) . Consider a finite sequence { Z k } of independent, random, matrices with common dimension d × d . Assume that each randommatrix satisfies E Z k = 0 and k Z k k R almost surely. Define σ = max ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k E [ Z k Z ∗ k ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k E [ Z ∗ k Z k ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) Then, for all t > , P (cid:2)(cid:13)(cid:13)(cid:13)X k Z k (cid:13)(cid:13)(cid:13) > t (cid:3) ( d + d ) exp (cid:18) − t / σ + Rt/ (cid:19) . The following lemma is a standard result, relying on semidefinite programming and theGrothendieck inequality to bound k M k ∞→ . Lemma 3.3.
Given a matrix M , one can efficiently approximate k M k ∞→ to within a constantfactor . of its true value.Proof. First note that the maximum value of x ⊤ M y over Boolean vectors x and y exactly themaximum of x ⊤ M y over vectors x and y with entries of magnitude at most 1. The Grothendieckinequality [Gro53] states thatmax k u i k , k v j k X i,j M i,j h u i , v j i K G max | x i | , | y j | X i,j M i,j x i y j u , . . . , u n , v , . . . , v n ∈ R d for some d , the rightmaximization is taken over numbers x , . . . x n , y , . . . , y n ∈ R , and K G is an absolute constantindependent of M , n , or d . It is known that K G π √ < . d = 2 n , and is clearly a relaxation ofthe case when d = 1 which is what we aim to bound. The Grothendieck inequality shows thatthe upper bound obtained from this relaxation cannot be more than a factor of K G larger than k M k ∞→ , so the solution to the SDP serves as a constant-factor approximation. Our main technical result is the following theorem about refutation of semi-random partitioned2-XOR.
Theorem 4.1 (Semi-random Partitioned 2-XOR Refutation) . Let < ε < / . Let ψ = ( V, ℓ, E, r ) be a semi-random partitioned 2-XOR instance in which | V | = n and | E | = m . Suppose ℓ = O (poly( n )) . Then there is an absolute constant K such that if m > max ( Kn √ ℓ (log n ) ε , ℓε ) we can with high probability efficiently certify that val ψ / ε. As an application of this result, we prove the following theorem about refutation of semi-random k -XOR as a corollary, which is just a more detailed version of Theorem 1.2. Theorem 4.2.
Let < ε < / . Let φ = ( V, E, r ) be a semi-random k -XOR instance in which | V | = n and | E | = m . Then there is an absolute constant K such that if m > Kn k/ (log n ) ε with high probability we can efficiently certify that val ψ / ε. Proof.
Suppose k is even. There is a natural relaxation of φ to a semi-random 2-XOR instance ψ = ( (cid:0) Vk/ (cid:1) , E ′ , r ′ ) as follows. For every k -set e ∈ E , break it into two ( k/ e , e which unionto e , and add the edge ( e , e ) to E ′ with sign r ′ ( e , e ) = r ( e ). Relax ψ to a partitioned 2-XORinstance ψ ′ with a single part. Then Theorem 4.1 with ℓ = 1 and vertex set size n k implies ψ ′ canbe (1 / ε )-refuted, which also (1 / ε )-refutes φ . We could alternatively refute semi-random2-XOR without using the full power of Theorem 4.1, which we prove later as Theorem 5.3.Suppose k is odd. We construct a partitioned 2-XOR instance ψ = ( (cid:0) V ( k − / (cid:1) , ℓ, E ′ , r ′ ) with ℓ = n as follows. For every e ∈ E , break it arbitrarily into a single vertex i , and two disjoint( k − / e and e such that i ∪ e ∪ e = e . Let E ′ be the multiset of all such ( i, ( e , e )).Define r ′ i ( e , e ) = r ( i ∪ e ∪ e ). Say x ∈ {± } n is the optimal assignment for φ . Let x ′ be the ± k − / V such that x ′ S = x S , and let y ′ = x . Thenval ψ > m { ( i, ( e , e ) ∈ E ′ | x ′ e x ′ e y ′ i r ′ i ( e , e ) = 1 }
12 1 m { ( i, ( e , e )) ∈ E ′ | x i ∪ e ∪ e r ( i ∪ e ∪ e ) = 1 } = 1 m { e ∈ E | x e r ( e ) = 1 } = val φ . Thus it suffices to (1 / ε )-refute the partitioned 2-XOR instance ψ , which we can do by applyingTheorem 4.1 with ℓ = n partitions, and vertex set size | V | = (cid:0) n ( k − / (cid:1) . The goal of this section is to prove Theorem 4.1, thus establishing Theorem 4.2. Our algorithmitself can be compactly described as follows:
Algorithm 5.1 (Refuting Partitioned k -XOR) .Given: A partitioned 2-XOR instance ψ = ( V, ℓ, E, r ) with n variables and m constraints. Output:
Return either “Not (1 / ε )-satisfiable” or “Unknown”. Operation:
1. Let ψ ′ = ( V ′ , E ′ , r ′ ) be an empty semi-random 2-XOR instance.2. While there exists i ∈ [ ℓ ] and v ∈ V such that there are at least Θ( ε − ) constraintsof the form ( i, ( v, u )) = C ∈ E , for each such constraint C , set V := V ∪ { ( i, v ) , u } and E ′ := E ′ + { (( i, v ) , u ) } and remove e from E .3. Let µ i ( u, v ) be the sum of r ( e ) over all (possibly zero) duplicates of e = ( i, ( u, v ))in E , and let t i be the number of e ∈ E with first coordinate i ∈ [ ℓ ]. If | E | > εm/ x v,v ′ ,y u,u ′ ∈ R n k x v,v ′ k = k y u,u ′ k =1 X ( v,u ) =( v ′ ,u ′ ) ∈ V ℓ X i =1 µ i ( v, u ) µ i ( v ′ , u ′ ) √ t i h x v,v ′ , y u,u ′ i = O ε | E | / √ l ! and if this does not hold, return “Unknown”.4. Let µ ′ (( i, v ) , u ) be the sum of r ( e ) over all (possibly zero) duplicates of e = (( i, v ) , u )in E ′ . If | E ′ | > εm/
2, verify with an SDP thatmax x i,v ,y u ∈ R | V ′| k x i,v k = k y u k =1 X ( i,v ) ∈ V ′ ∩ ([ ℓ ] × V ) X u ∈ V ′ ∩ V µ ′ (( i, v ) , u ) h x i,v , y u i = O ( ε | E ′ | )and if this does not hold, return “Unknown”.5. If both verifications passed, return “Not (1 / ε )-satisfiable”.13 .1 Decomposing Semi-random Partitioned 2-XOR For a semi-random partitioned 2-XOR problem ψ = ( V, ℓ, E, r ), let deg i ( v ) be the number of( i, e ) ∈ E such that v ∈ E . The main idea of the proof of Theorem 5.2 is to split the constraints ofa general semi-random partitioned 2-XOR instance between two instances ψ and ψ . The instance ψ will satisfy deg i ( v ) d = O ( ε − ) for all i ∈ [ ℓ ] and all v ∈ V . In the instance ψ , for every( i, e ) ∈ E is adjacent to some vertex v with degree at least d within its partition. We use a degree-bounded semi-random partitioned 2-XOR refutation to bound ψ , and then relax the ψ into asemi-random 2-XOR instance and refute this as well. We establish the following two theorems withthe aim of refuting ψ and ψ above. Theorem 5.2 ( d -bounded Semi-random Partitioned 2-XOR Refutation) . Let < ε < / . Let ψ = ( V, ℓ, E, r ) be a semi-random partitioned 2-XOR instance with | V | = n , ℓ = O (poly( n )) , and | E | = m . Suppose deg i ( v ) d = O (poly( ε − , log n )) for all i ∈ [ ℓ ] and v ∈ V . Then there is anabsolute constant K such that if m > max ( Kdn √ ℓ (log n ) ε , ℓε ) we can with high probability efficiently certify that val φ / ε. We will also use the following analysis of SDP based refutation whose proof we will laterreproduce for completeness. This idea was used in the context of refutation for the first time in thework of Feige [Fei07].
Theorem 5.3 (Semi-random 2-XOR refutation) . Let < ε < / . Let φ = ( V, E, r ) be a semi-random 2-XOR instance with | V | = n and | E | = m . Then there is an absolute constant K suchthat for m > Knε we can with high probability efficiently certify that val φ / ε. The proofs of Theorem 5.2 and Theorem 5.3 will be given later in this section. For now, let ustake them as facts and use them to prove Theorem 4.1. d -bounded and -XOR cases for general partitioned -XOR refu-tation We start by iteratively defining two multisets E and E such that E + E = E . Let E initially beempty, and let E initially be equal to E . For i ∈ [ ℓ ] and v ∈ V , let S ( i, v ) = { u ∈ V | ( i, { v, u } ) ∈ E } . Let C be a fixed constant, whose exact value will be chosen later. While there exist i and v such that | S ( i, v ) | > Cε − , remove everything in S ( i, v ) from E and add them all to E instead.Repeat this process until no such i and v exist. We have now split E into two separate semi-randompartitioned 2-XOR instances (defined by E and E ) over the same vertex set V , each of whichwe will refute separately. Let ψ = ( V, ℓ, E , r ) be the first instance and ψ = ( V, ℓ, E , r ) be thesecond. 14 emma 5.4. If | E | > εm/ , we can (1 / ε/ -refute ψ . Lemma 5.5. If | E | > εm/ , we can (1 / ε/ -refute ψ . The proofs of Lemmas 5.4 & 5.5 (which we give later in this subsection) respectively describeSteps 3 & 4 of Algorithm 5.1. Using the above two lemmas, we can prove Theorem 4.1.
Proof of Theorem 4.1.
Begin our procedure by pruning as above to obtain the sets E and E . Ifthe original instance ψ has a (1 / ε )-satisfying solution ( x, y ), then across both instances ψ and ψ we can satisfy at least (1 / ε ) m constraints by choosing the assignment ( x, y ) for each instance.Let s be the maximum number of constraints we can simultaneously satisfy in ψ , and let s j be themaximum number of constraints we can simultaneously satisfy in ψ j . We give three cases, in eachone bounding s s + s by (1 / ε ) m to complete the proof.Suppose both | E | , | E | > εm/
2. Then we may (1 / ε/
2) refute both instances by Lemma 5.4,and Lemma 5.5, showing that s + s (1 / ε/ | E | + | E | ) (1 / ε ) m. Now suppose exactly one of | E | and | E | is at least εm/
2, and the other has size below εm/ E . Refute the first instance, and note that | s | | E | εm/
2. Then s + s (1 / ε/ | E | + εm/ (1 / ε ) m. There is no case when both of the | E j | are small, since they would comprise of at most εm totalconstraints.All that remains is to prove Lemma 5.4 and Lemma 5.5. Proof of Lemma 5.4.
Within each of the ℓ partitions T i of ψ , each of the n vertices is adjacent toat most Cε − edges because of the way we pruned E . Then in the style of Theorem 5.2 we have adegree bound of d = Cε − . With a large enough constant in our lower bound on m , for any fixedconstant R we can force | E | > εm/ > RCn √ ℓ (log n ) ε = Rdn √ ℓ (log n ) ( ε/ . Therefore by Theorem 5.2 with high probability we may (1 / ε/ d -bounded partitioned2-XOR instance ψ . Proof of Lemma 5.5.
We reduce refutation of E to refutation of (non-partitioned) semi-random2-XOR. Let m ′ = | E | . Recall that we constructed E by repeatedly adding sets S ( i, v ) of sizeat least Kε − , where every element in S ( i, v ) was of the form ( i, { v, u } ). Let X = { ( i, v ) | S ( i, v )was added to E during its construction } . Since each set S ( i, v ) is disjoint from all other such sets,there are m ′ constraints in the S ( i, v ) in total, and each | S ( i, v ) | > Cε − , it follows that | X | ε m ′ C .We a bipartite semi-random 2-XOR instance φ = ( X ∪ V, E ′ , r ′ ) on the bipartitioned vertex set X ∪ V as follows. Let E ′ = { (( i, v ) , u ) | ( i, { v, u } ) was originally added to E via S ( i, v ) } .
15e can think the constraint graph of φ intuitively follows: For each edge e ∈ T i there is some( i, v ) ∈ X such that v ∈ e . Then an edge e ∈ T i is transformed into an edge between ( i, v ) and thevertex u ∈ e which is not v .We inherit the random sign function r ′ : E ′ → {± } from r by setting r ′ (( i, v ) , u ) = r i ( v, u ).The distribution of r ′ is uniform over all functions E ′ → {± } , since r ′ is just r carried over to E ′ via an isomorphism of sets.We show now that refuting the 2-XOR instance φ is a relaxation of refuting the partitioned2-XOR instance with constraint set E , so it suffices to refute the former. Say x, y is an maximallysatisfying assignment for the partitioned 2-XOR instance with constraint set E . Let x ′ be a vectorin {± } X ∪ V defined as follows. Let x ′ ( i,v ) = y i x v for ( i, v ) ∈ X and let x ′ v = x v for v ∈ V . Then for e = ( v, u ) we have x e y i r i ( e ) = 1 = ⇒ x v y i x u r ′ i,v ( u ) = 1 = ⇒ x ′ v x ′ ( i,v ) r ′ i,v ( u ) = 1so the maximally satisfying assignment to E ′ will satisfy at least as many constraints as themaximally satisfying assignment to E .Let n ′ = | X | + | V | , the number of variables in φ . Then n ′ ε m ′ /C + n ε m ′ /C since ε m ′ /C > ε m/ (2 C ) > Kn √ ℓ (log n ) Cε > n for sufficiently large n . By choosing C large enough, wecan force m ′ > Kn ′ ( ε/ for any fixed constant K , forcing the conditions of Theorem 5.3 to holdthereby allowing for (1 / ε/ Let us now reproduce the proof of the needed 2-XOR refutation and complete the case handled byTheorem 5.3.
Proof of Theorem 5.3.
Extend the domain of r to all of V , so that previously undefined valuesare now 0. Let there be a canonical pair representing each edge, so that at most one of r ( u, v ) and r ( v, u ) is nonzero. Define M ∈ R V × V by M ( v, u ) = r ( v, u ). Then(2 val φ − m = max x ∈{± } V x ⊤ M x k M k ∞→ so it suffices to certify that k M k ∞→ εm. It follows from the Lemma 3.3 that we can bound this maximum of x ⊤ M y over Boolean vectors x and y to within a constant factor of its true value using an SDP. Therefore, for refutation it sufficesto show that the maximal value of x ⊤ M y over Boolean vectors is actually below
Cεm with highprobability for some sufficiently small constant C .Fix x, y ∈ {± } n . Set t = Cεm . By Bernstein’s inequality,P X ( u,v ) ∈ e x u y v r ( u, v ) > t exp (cid:18) − t / m + t/ (cid:19) exp( − C ε m ) exp( − C Kn )Where the last inequality follows since m > Kn/ε . By choosing large enough K we can forcethis value below 1 / ( p ( n )2 n ) for a polynomial p ( n ) of arbitrarily high degree. Then taking a unionbound over all 2 n choices of fixed x, y ∈ {± } n allows us to certify x ⊤ M y Cεm to any desiredpolynomial probability of success. 16 .4 The d -Bounded Case: Proof of Theorem 5.2 The remainder of this section (and the paper) is devoted to proving Theorem 5.2 on the refutationof degree-bounded semi-random partitioned 2-XOR.Define the multiset T i = { e | ( i, e ) ∈ E } , containing one copy of e for each copy of ( i, e ) in E .Let t i = | T i | . We assume that each T i is nonempty, as lowering ℓ only makes an instance easier torefute.Define a function Φ : {± } n → R as follows.Φ( x ) = ℓ X i =1 √ t i X e ∈ T i x e r i ( e ) The following lemma shows that when val ψ has a large value, Φ( x ) can take on a large value aswell. Lemma 5.6. If val ψ > / ε , then there is x ∈ {± } V such that Φ( x ) > ε m / ℓ / Proof.
Let ( x, y ) be a maximally satisfying assignment to ψ . Let s i be the number of edges in T i which are satisfied by ( x, y ). Note that P i s i > (1 / ε ) m , P i t i = m , and P e ∈ T i x e r i ( e ) y i = 2 s i − t i .We use the fact that every y i = 1 to showΦ( x ) = ℓ X i =1 √ t i X e ∈ T i x e r i ( e ) = ℓ X i =1 √ t i X e ∈ T i x e r i ( e ) y i = ℓ X i =1 (2 s i − t i ) √ t i > (cid:16)P ℓi =1 s i − t i (cid:17) P ℓi =1 √ t i > (cid:0) / ε ) m − m (cid:1) √ ℓm = 4 ε m / ℓ / where the first inequality follows from Cauchy-Schwarz applied to the sequences u i = (2 s i − t i ) /t / i and v i = t / i , and the second inequality follows from Cauchy-Schwarz applied to P i √ t i , as wellas our bounds on P i s i and P i t i .So to certify val φ / ε , it suffices to certify Φ( x ) ε m / ℓ / for all x .Notice that by focusing on bounding Φ( x ), we have effectively removed y from consideration.Write Φ( x ) = ℓ X i =1 √ t i X ( e,e ′ ) ∈ ( Ti ) x e x e ′ r i ( e ) r i ( e ) + ℓ X i =1 √ t i X e ∈ T i ( x e ) r i ( e ) and let Φ ( x ) be the first term above and Φ ( x ) be the second. We bound Φ( x ) by bounding thetwo terms individually. Lemma 5.7.
We have the upper bound Φ ( x ) ε m / ℓ / . roof. See that every term in the inner summation in Φ ( x ) is 1, since r i ( e ) , x e ∈ {± } . Thus theentire inner summation is t i , meaningΦ ( x ) = ℓ X i =1 √ t i vuut ℓ ℓ X i =1 t i = √ ℓm where the inequality follows from Cauchy-Schwarz. Since ℓ ε m by our lower bound on m , wesee √ ℓm ε m / ℓ / , completing the proof.Lemma 5.8 is always true, so requires no certification. Thus, if we can prove the followinglemma, we will be done. Lemma 5.8.
With high probability, we can certify that Φ ( x ) ε m / ℓ / for all x ∈ {± } n . The rest of this section is dedicated to proving Lemma 5.8. For i ∈ [ ℓ ], define the randomfunction µ i : (cid:0) V (cid:1) → Z as the sum of r i ( e ) over all copies of ( i, e ) ∈ E . Then define the matrix M ∈ R V × V by M ( v, v ′ ; u, u ′ ) = (P ℓi =1 1 √ t i µ i ( v, u ) µ i ( v ′ , u ′ ) when ( v, u ) = ( v ′ , u ′ ), and0 otherwise.Let x ⊗ be the vector with ( x ⊗ ) u,v = x u x v . By tracing definitions, we see thatΦ ( x ) = ( x ⊗ ) ⊤ M ( x ⊗ ) k M k ∞→ so if we can certify that k M k ∞→ ε m / √ ℓ we are done. By Lemma 3.3, we can efficientlyapproximate this value within a factor
3, so it suffices to just show that k M k ∞→ is at most ε m / √ ℓ .We start by assigning weights γ ( v, v ′ ) to all pairs ( v, v ′ ) ∈ V . Define γ ( v, v ′ ) = ℓ X i =1 deg i ( v ) deg i ( v ′ ) t i . (5.1)We refer to γ ( v, v ′ ) as the butterfly degree of the pair ( v, v ′ ). We can think of the butterfly degreeas counting (weighted) diagrams of the form in Figure 1, where a diagram in partition set i is givenweight 1 /t i . Intuitively, γ ( v, v ′ ) will be high when v and v ′ have relatively high degrees withinrelatively small T i . The next lemma shows that the total of the butterfly degrees of all pairs is nottoo large. Lemma 5.9.
We have X v,v ′ ∈ [ n ] γ ( v, v ′ ) = 4 m roof. Indeed, X v,v ′ ∈ [ n ] γ ( v, v ′ ) = ℓ X i =1 t i X v,v ′ ∈ V deg i ( v ) deg i ( v ′ ) = ℓ X i =1 t i X v ∈ V deg i ( v ) ! = ℓ X i =1 t i (2 t i ) = 4 m . We now define a partition S , S , . . . , S log n of V by breaking pairs ( v, v ′ ) into different “butterflydegree classes”. We define two numbers α, β ∈ R as follows: α = Cd ℓ (log n ) ε m , β = (cid:18) mα (cid:19) / log n (5.2)Where C is some large enough absolute constant, with the precise meaning of “large enough” tobe determined later.We define the sets S j as follows. S = (cid:8) ( v, v ′ ) ∈ V | γ ( v, v ′ ) α (cid:9) and S j = (cid:8) ( v, v ′ ) ∈ V | αβ j − γ ( v, v ′ ) αβ j (cid:9) for j > γ ( v, v ) = e O (1) for each pair v, v ′ , and S = V . We can think of S as containing all pairs whichbehave pseudorandomly, while S j for larger j contain pairs whose edge mass is larger and moreheavily correlated.In order for this partition to be legitimate, we need to establish that β > { S j } actually include all elements of V . We will show now that β is bounded on both sides byabsolute constants. Lemma 5.10.
The parameter β defined in (5.2) satisfies β = O (1) . Proof.
We first establish the upper bound. Since each of the n elements of V is in at most d pairsin each of the ℓ partitions, we know m ℓdn = O (poly( n )). Then expanding α and β we have β = (cid:18) ε m Cd ℓ (log n ) (cid:19) / log n O (cid:0) ℓn (cid:1) / log n = O (poly( n )) / log n = O (1) . Next we establish the lower bound of 1. β = (cid:18) ε m Cd ℓ (log n ) (cid:19) / log n > (cid:18) ε mCd (log n ) (cid:19) / log n > / log n = 1 . where the first inequality holds since we assumed ℓ ε m via our lower bound on m , and thesecond inequality holds since d = O (poly( ε − , log n )).By Lemma 5.9 and nonnegativity of γ , we know γ ( v, v ′ ) m = αβ log n always, so the S j doindeed cover all possible cases for γ ( v, v ′ ) and thus form a partition.19 emma 5.11. | S | n , and for j > , | S j | mαβ j − . Proof.
The first bound follows as S ⊆ V . For the second bound see that | S j | αβ j − X ( v,v ′ ) ∈ S j γ ( v, v ′ ) X v,v ′ ∈ V γ ( v, v ′ ) = 4 m where the last inequality follows from Lemma 5.9.The main idea of our refutation is as follows. A pair ( v, v ′ ) of high butterfly degree will bumpup the value of Φ . However, Lemma 5.11 shows that there cannot be too many pairs contributingtoo large of a value. To make this more precise, we will break M down into a sum of (1 + log n ) blocks M j,k indexed by 0 j, k log n . Define M j,k as just the restriction of M to rows in S j andcolumns in S k . Then clearly x ⊤ M y = P j,k x ⊤| S j M j,k y | S k , so k M k ∞→ X j,k log n k M j,k k ∞→ . We will bound each k M j,k k ∞→ individually via the inequality k M j,k k ∞→ = max x,y Boolean x ⊤ M y q | S j | · | S k | · k M j,k k . Intuitively, a similar bound will not work on k M k ∞→ directly because much of its mass can beunevenly distributed, causing the eigenvector associated to its maximal eigenvalue to look very farfrom a Boolean vector which results in a loose bound. However, we might expect a spectral boundon the M j,k will work since by restricting to rows/columns in certain butterfly degree classes S j and S k , we cause the vectors maximizing x ⊤ M y to more closely resemble Boolean vectors.As Lemma 5.11 gives a bound on the size of the sets S j and S k , what remains is to bound k M j,k k . Lemma 5.12.
For all j, k log n , we have k M j,k k = O (cid:18) d q αβ max( j,k ) (log n ) (cid:19) . with high probability. The key ingredient in the proof of Lemma 5.12 is the rectangular matrix Bernstein inequality,stated earlier as Theorem 3.2. To apply it, we will write M j,k as a sum of independent, zero-meanmatrices B i,j,k for 1 i ℓ , and provide upper bounds on both k B i,j,k k and the variance term σ from the statement of the rectangular matrix Bernstein inequality. We begin by defining thematrix B i,j,k ∈ R S j × S k as B i,j,k ( v, v ′ ; u, u ′ ) = ( √ t i µ i ( v, u ) µ i ( v ′ , u ′ ) when ( v, u ) = ( v ′ , u ′ ), and0 otherwise. (5.3)It follows from definitions that P ℓi =1 B i,j,k = M j,k . We now establish a uniform bound on thespectral norm of B i,j,k . 20 emma 5.13. For all j, k log n and i n , k B i,j,k k d q αβ max( j,k ) Proof.
Fix v, v ′ ∈ V . Let w i = deg i ( v ) deg i ( v ′ ). The ℓ norm of the row corresponding to ( v, v ′ ) in B i,j,k is at most w i √ t i , since each pair of edges e, e ′ ∈ T i with v ∈ e and v ′ ∈ e ′ contributes at most √ t i to it, by the definition of B i,j,k . Since the instance is d -bounded we know w i d , so d √ w i > w i √ t i (cid:18) d √ w i (cid:19) · w i √ t i = d p w i /t d vuut ℓ X i =1 w i /t i = d p γ ( v, v ′ ) d p αβ j where the last inequality follows since ( v, v ′ ) ∈ S j . This shows the ℓ norm of rows of B i,j,k arebounded by d p αβ j , and an analogous argument works to show that the max ℓ norm of any columnis bounded by d p αβ k . Then by Lemma 3.1 we are done.All that remains is to bound the variance term in the statement of Theorem 3.2. We prove thefollowing lemma. Lemma 5.14.
For j, k log n , we have max ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ X i =1 E [ B i,j,k B ⊤ i,j,k ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ X i =1 E [ B ⊤ i,j,k B i,j,k ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) αβ max( j,k ) Proof.
Let X j,k = P ℓi =1 E [ B i,j,k B ⊤ i,j,k ]. Because B i,j,k = B ⊤ i,k,j , the value we would like to bound ismax {k X j,k k , k X k,j k} . Then it suffices to show that k X j,k k αβ j .We will first show that X j,k ( v, v ′ ; u, u ′ ) = 0 implies { v, v ′ } = { u, u ′ } . We expand the definitionto find X j,k ( v, v ′ ; u, u ′ ) = ℓ X i =1 X w,w ′ E [ B i,j,k ( v, v ′ ; w, w ′ ) B i,j,k ( u, u ′ ; w, w ′ )]= ℓ X i =1 t i · X ( w,w ′ ) ∈ S k ( v,w ) =( v ′ ,w ′ )( u,w ) =( u ′ ,w ′ ) E [ µ i ( v, w ) µ i ( v ′ , w ′ ) µ i ( u, w ) µ i ( u ′ , w ′ )]For any e, e ′ ∈ T i either µ i ( e ) and µ i ( e ′ ) are identical or µ i ( e ) and µ i ( e ′ ) are independent withzero-mean. A term is nonzero precisely when the four µ ( − ) factors can be partitioned into twoequal pairs. We cannot pair factors 1 and 2 since { v, w } 6 = { v ′ , w ′ } by the condition on indices weare summing over.Suppose factors 1 and 3 are equal, and additionally factors 2 and 4 are equal. Then we can seedirectly that v = u and v ′ = u ′ .Suppose factors 1 and 4 are equal, and additionally factors 2 and 3 are equal. Then either (1) v = u ′ and w = w ′ , or (2) v = w ′ and w = u ′ . By looking at factors 2 and 3, case (1) implies v ′ = u ,meaning ( v, v ′ ) = ( u ′ , u ). Again looking at factors 2 and 3, case (2) implies { v, v ′ } = { u, u ′ } .Thus by considering all cases, we have determined that X j,k ( v, v ′ ; u, u ′ ) is only ever nonzerowhen { v, v ′ } = { u, u ′ } . Let D i ( v, w ) be the number of copies of edge ( v, w ) in partition set i . Then21 [ µ i ( v, w ) ] = D i ( v, w ) since it is a sum of D i ( v, w ) independent Rademacher random variables. Inthe case v = u and v ′ = u ′ , we see X j,k ( v, v ′ ; v, v ′ ) = ℓ X i =1 t i · X ( w,w ′ ) ∈ S k ( v,w ) =( v ′ ,w ′ ) E [ µ i ( v, w ) µ i ( v ′ , w ′ ) ]= ℓ X i =1 t i · X ( w,w ′ ) ∈ S k ( v,w ) =( v ′ ,w ′ ) D i ( v, w ) D i ( v ′ , w ′ ) ℓ X i =1 deg i ( v ) deg i ( v ′ ) t i = γ ( v, v ′ ) αβ j where the last inequality follows since ( v, v ′ ) ∈ S j . When v = u ′ and v ′ = u , we see X j,k ( v, v ′ ; v ′ , v ) = ℓ X i =1 t i · X ( w,w ′ ) ∈ S k ( v,w ) =( v ′ ,w ′ )( v ′ ,w ) =( v,w ′ ) E [ µ i ( v, w ) µ i ( v ′ , w ′ ) µ i ( v ′ , w ) µ i ( v, w ′ )] . By inspecting the four factors in the expectation, and considering the two cases where they canbe broken into two equal pairs, we see that the above is at most ℓ X i =1 t i · X ( w,w ′ ) ∈ V E [ µ i ( v, w ) µ i ( v ′ , w ′ ) ]= γ ( v, v ′ ) αβ j where the last equality is because ( v, v ′ ) ∈ S j . Then each row and column of X j,k has at most twononzero entries, whose total sum is bounded by 2 γ ( v, v ′ ) αβ j . Then Lemma 3.1 applies to show k X j,k k αβ j , completing the proof.We now have enough information to apply the rectangular matrix Bernstein inequality to thedecomposition M j,k = P ℓi =1 B i,j,k . By Theorem 3.2, Lemma 5.13, and Lemma 5.14, we can setsome t = Θ (cid:16) d p αβ max( i,j ) (log n ) (cid:17) so thatP [ k M j,k k > t ] ( | S j | + | S k | ) exp − t / αβ max( j,k ) + d p αβ max( j,k ) t/ ! n − Θ(log n ) p ( n )for an arbitrarily high degree polynomial p . This completes the proof of Lemma 5.12. Since thereare only (1 + log n ) different M j,k the probability that k M j,k k = O ( αβ max( i,j ) ) for all of them isat least 1 − p ( n ) for a polynomial p of arbitrarily high degree. We now show that in the highlyprobable case that this bound holds for all M j,k , Φ ( x ) ε m / √ ℓ for all x ∈ {± } n .22e have the inequality k M k ∞→ = X j,k log n k M j,k k ∞→ max j,k log n k M j,k k ∞→ (log n ) (5.4) max j,k log n q | S j | · | S k | · k M j,k k (log n ) . (5.5)We show that, no matter the choice of j and k , the above bound is sufficiently small. Since ourupper bound on | S j | from Lemma 5.11 depends on whether j = 0 or not, we break into three cases.We only consider the case when j k , since k M j,k k ∞→ = k M k,j k ∞→ .When the right side of Inequality (5.5) is maximized at j = k = 0, we have2 p | S | · | S | · k M , k (log n ) n k M , k (log n ) (by Lemma 5.11)= O (cid:0) dn √ α (log n ) (cid:1) . (by Lemma 5.12)When the right side of Inequality (5.5) is maximized at j = 0 and k >
1, we have2 p | S | · | S k | · k M ,k k (log n ) n r mαβ k − k M ,k k (log n ) (by Lemma 5.11)= O (cid:16) n p βd √ m (log n ) (cid:17) (by Lemma 5.12)= O (cid:0) dn √ m (log n ) (cid:1) . (by Lemma 5.10)When the right side of Inequality (5.5) is maximized at 1 j k , we have2 q | S j | · | S k | · k M j,k k (log n ) mαβ ( j + k − / k M j,k k (log n ) (by Lemma 5.11)= O (cid:18) dm (log n ) √ αβ ( j − / (cid:19) (by Lemma 5.12)= O (cid:18) dm (log n ) √ α (cid:19) . (by Lemma 5.10)Recall that we defined α = Cd ℓ (log n ) ε m for some large enough absolute constant C . The thirdupper bound can be forced below any fixed constant fraction of ε m / √ ℓ by increasing the constant C in the definition of R . Additionally, the first two upper bounds can be forced below any fixedconstant fraction of ε m / √ ℓ by increasing the constant K in the lower bound on m from the statementof Theorem 5.2. This completes the proof of Theorem 5.2. References [AAT05] Mikhail Alekhnovich, Sanjeev Arora, and Iannis Tourlakis,
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