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22
Gowers uniformity, influence of variables, and PCPs
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimensiond uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We pro ..."
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Cited by 65 (2 self)
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Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimensiond uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We prove the following inverse theorem. Write G = G1 × · · · × Gn as a product of groups. If a bounded balanced function f: G1 × · · · Gn → C is such that U d (f) ≥ ε, then one of the coordinates of f has influence at least ε/2 O(d). Other inverse theorems are known [Gow98, Gow01, GT05, Sam05], and U 3 is especially well understood, but the properties of functions f with large U d (f), d ≥ 4, are not yet well characterized. The dimensiond Gowers inner product 〈{fS} 〉 U d of a collection {fS} S⊆[d] of functions is a related measure of pseudorandomness. The definition is such that if all the functions fS are equal to the same fixed function f, then 〈{fS} 〉 U d = U d (f). We prove that if fS: G1 × · · · × Gn → C is a collection of bounded functions such that 〈{fS} 〉 U d  ≥ ε and at least one of the fS is balanced, then there is a variable that has influence at least ε 2 /2 O(d) for at least four functions in the collection. Finally, we relate the acceptance probability of the “hypergraph longcode test ” proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce the following result: if the Unique Games Conjecture is true, then for every q ≥ 3 there is a PCP characterization of NP where the verifier makes q queries, has almost perfect completeness, and soundness at most 2q/2 q. For infinitely many q, the soundness is (q + 1)/2 q, which might be a tight result. Two applications of this results are that, assuming that the unique games conjecture is true, it is hard to approximate Max kCSP within a factor 2k/2 k ((k + 1)/2 k for infinitely many k), and it is hard to approximate Independent Set in graphs of degree D within a factor (log D) O(1) /D. 1
CSP Gaps and Reductions in the Lasserre Hierarchy
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 104 (2008)
, 2008
"... We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck [25] recently showed the first integrality gaps for these problems, showing that for MAX kXOR, the ratio of the SDP optimum to the integer optimum may be as l ..."
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Cited by 42 (5 self)
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We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck [25] recently showed the first integrality gaps for these problems, showing that for MAX kXOR, the ratio of the SDP optimum to the integer optimum may be as large as 2 even after Ω(n) rounds of the Lasserre hierarchy. We show that for the general MAX kCSP problem over binary domain, the ratio of SDP optimum to the value achieved by the optimal assignment, can be as large as 2 k /2k − ɛ even after Ω(n) rounds of the Lasserre hierarchy. For alphabet size q which is a prime, we give a lower bound of q k /q(q − 1)k − ɛ for Ω(n) rounds. The method of proof also gives optimal integrality gaps for a predicate chosen at random. We also explore how to translate gaps for CSP into integrality gaps for other problems using reductions, and establish SDP gaps for Maximum Independent Set, Approximate Graph Coloring, Chromatic Number and Minimum Vertex Cover. For Independent Set and Chromatic Number, we show integrality gaps of n/2 O( √ log nlog log n) even after 2
Approximation Resistant Predicates From Pairwise Independence
, 2008
"... We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwis ..."
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Cited by 32 (5 self)
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We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwise independent distribution over [q] k whose support is contained in the set of satisfying assignments to P. Using constructions of pairwise independent distributions this result implies that • For general k ≥ 3 and q ≥ 2, theMAX kCSPq problem is UGhard to approximate within O(kq 2)/q k + ɛ. • For the special case of q =2, i.e., boolean variables, we can sharpen this bound to (k + O(k 0.525))/2 k + ɛ, improving upon the best previous bound of 2k/2 k +ɛ (Samorodnitsky and Trevisan, STOC’06) by essentially a factor 2. • Finally, again for q =2, assuming that the famous Hadamard Conjecture is true, this can be improved even further, and the O(k 0.525) term can be replaced by the constant 4. 1
Sound 3query PCPPs are long
, 2008
"... We initiate the study of the tradeoff between the length of a probabilistically checkable proof of proximity (PCPP) and the maximal soundness that can be guaranteed by a 3query verifier with oracle access to the proof. Our main observation is that a verifier limited to querying a short proof cannot ..."
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Cited by 10 (3 self)
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We initiate the study of the tradeoff between the length of a probabilistically checkable proof of proximity (PCPP) and the maximal soundness that can be guaranteed by a 3query verifier with oracle access to the proof. Our main observation is that a verifier limited to querying a short proof cannot obtain the same soundness as that obtained by a verifier querying a long proof. Moreover, we quantify the soundness deficiency as a function of the prooflength and show that any verifier obtaining “best possible” soundness must query an exponentially long proof. In terms of techniques, we focus on the special class of inspective verifiers that read at most 2 proofbits per invocation. For such verifiers we prove exponential lengthsoundness tradeoffs that are later on used to imply our main results for the case of general (i.e., not necessarily inspective) verifiers. To prove the exponential tradeoff for inspective verifiers we show a connection between PCPP proof length and propertytesting query complexity, that may be of independent interest. The connection is that any linear property that can be verified with proofs of length ℓ by linear inspective verifiers must be testable with query complexity ≈ log ℓ.
Constraint Satisfaction over a NonBoolean Domain: Approximation Algorithms and UniqueGames Hardness
 In Proceedings of APPROX 2008
, 2008
"... We study the approximability of the Max kCSP problem over nonboolean domains, more specifically over {0, 1,..., q − 1} for some integer q. We extend the techniques of Samorodnitsky and Trevisan [18] to obtain a UGC hardness result when q is a prime. More precisely, assuming the Unique Games Conjec ..."
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Cited by 9 (1 self)
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We study the approximability of the Max kCSP problem over nonboolean domains, more specifically over {0, 1,..., q − 1} for some integer q. We extend the techniques of Samorodnitsky and Trevisan [18] to obtain a UGC hardness result when q is a prime. More precisely, assuming the Unique Games Conjecture, we show that it is NPhard to approximate the problem to a ratio greater than q2k/qk. Independent of this work, Austrin and Mossel [2] obtain a more general UGC hardness result using entirely different techniques. We also obtain an approximation algorithm that achieves a ratio of C(q) · k/qk for some constant C(q) depending only on q. Except for constant factors depending on q, the algorithm and the UGC hardness result have the same dependence on the arity k. It has been pointed out to us [14] that a similar approximation ratio can be obtained by reducing the nonboolean case to a boolean CSP, and appealing to the CMM algorithm [3]. As a subroutine, we design a constant factor(depending on q) approximation algorithm for the problem of maximizing a semidefinite quadratic form, where the variables are constrained to take values on the corners of the qdimensional simplex. This result generalizes an algorithm of Nesterov [15] for maximizing semidefinite quadratic forms where the variables take {−1, 1} values.
On the Usefulness of Predicates
, 2012
"... Motivated by the pervasiveness of strong inapproximability results for MaxCSPs, we introduce a relaxed notion of an approximate solution of a MaxCSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly realvalued) c ..."
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Cited by 9 (1 self)
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Motivated by the pervasiveness of strong inapproximability results for MaxCSPs, we introduce a relaxed notion of an approximate solution of a MaxCSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly realvalued) constraints, and then only needs to satisfy as many of the new constraints as possible. To be more precise, we introduce the following notion of a predicate P being useful for a (realvalued) objective Q: given an almost satisfiable MaxP instance, there is an algorithm that beats a random assignment on the corresponding MaxQ instance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a MaxCSP with predicate P is exactly the same as saying that P is useful for P itself. We say that P is useless if it is not useful for any Q. This turns out to be equivalent to the following pseudorandomness property: given an almost satisfiable instance of MaxP it is hard to find an assignment such that the induced distribution on kbit strings defined by the instance is not essentially uniform. Under the Unique Games Conjecture, we give a complete and simple characterization of useful MaxCSPs defined by a predicate: such a MaxCSP is useless if and only if there is a pairwise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain MaxCSPs.
SDP Gaps from Pairwise Independence
"... This work considers the problem of approximating fixed predicate constraint satisfaction problems (MAX kCSP(P)). We show that if the set of assignments accepted by P contains the support of a balanced pairwise independent distribution over the domain of the inputs, then such a problem on n variable ..."
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Cited by 7 (0 self)
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This work considers the problem of approximating fixed predicate constraint satisfaction problems (MAX kCSP(P)). We show that if the set of assignments accepted by P contains the support of a balanced pairwise independent distribution over the domain of the inputs, then such a problem on n variables cannot be approximated better than the trivial (random) approximation, even after augmenting the natural semidefinite relaxation with Ω(n) levels of the SheraliAdams hierarchy. It was recently shown [3] that under the Unique Game Conjecture, CSPs for predicates satisfying this condition cannot be approximated better than the trivial approximation. Our results can be viewed as an unconditional analogue of this result in a restricted computational model. We also introduce a new generalization of techniques to define consistent “local distributions” over partial assignments to variables in the problem, which is often the crux of proving lower bounds for such hierarchies.
Optimal SheraliAdams Gaps from Pairwise Independence
"... This work considers the problem of approximating fixed predicate constraint satisfaction problems (MAX kCSP(P)). We show that if the set of assignments accepted by P contains the support of a balanced pairwise independent distribution over the domain of the inputs, then such a problem on n variable ..."
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Cited by 6 (1 self)
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This work considers the problem of approximating fixed predicate constraint satisfaction problems (MAX kCSP(P)). We show that if the set of assignments accepted by P contains the support of a balanced pairwise independent distribution over the domain of the inputs, then such a problem on n variables cannot be approximated better than the trivial (random) approximation, even using Ω(n) levels of the SheraliAdams LP hierarchy. It was recently shown [3] that under the Unique Game Conjecture, CSPs for predicates satisfying this condition cannot be approximated better than the trivial approximation. Our results can be viewed as an unconditional analogue of this result in the restricted computational model defined by the SheraliAdams hierarchy. We also introduce a new generalization of techniques to define consistent “local distributions ” over partial assignments to variables in the problem, which is often the crux of proving lower bounds for such hierarchies.
BlackBox Reductions in Mechanism Design
, 2011
"... A central question in algorithmic mechanism design is to understand the additional difficulty introduced by truthfulness requirements in the design of approximation algorithms for social welfare maximization. In this paper, by studying the problem of singleparameter combinatorial auctions, we obtai ..."
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A central question in algorithmic mechanism design is to understand the additional difficulty introduced by truthfulness requirements in the design of approximation algorithms for social welfare maximization. In this paper, by studying the problem of singleparameter combinatorial auctions, we obtain the first blackbox reduction that converts any approximation algorithm to a truthful mechanism with essentially the same approximation factor in a priorfree setting. In fact, our reduction works for the more general class of symmetric singleparameter problems. Here, a problem is symmetric if its allocation space is closed under permutations. As extensions, we also take an initial step towards exploring the power of blackbox reductions for general singleparameter and multiparameter problems by showing several positive and negative results. We believe that the algorithmic and game theoretic insights gained from our approach will help better understand the tradeoff between approximability and the incentive compatibility.