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Approximation algorithms for unique games
 In FOCS ’05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
"... Abstract: A unique game is a type of constraint satisfaction problem with two variables per constraint. The value of a unique game is the fraction of the constraints satisfied by an optimal solution. Khot (STOC’02) conjectured that for arbitrarily small γ,ε> 0 it is NPhard to distinguish games of ..."
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Abstract: A unique game is a type of constraint satisfaction problem with two variables per constraint. The value of a unique game is the fraction of the constraints satisfied by an optimal solution. Khot (STOC’02) conjectured that for arbitrarily small γ,ε> 0 it is NPhard to distinguish games of value smaller than γ from games of value larger than 1 − ε. Several recent inapproximability results rely on Khot’s conjecture. Considering the case of subconstant ε, Khot (STOC’02) analyzes an algorithm based on semidefinite programming that satisfies a constant fraction of the constraints in unique games of value 1 − O(k−10 · (logk) −5), where k is the size of the domain of the variables. We present a polynomial time algorithm based on semidefinite programming that, given a unique game of value 1−O(1/logn), satisfies a constant fraction of the constraints, where n is the number of variables. This is an improvement over Khot’s algorithm if the domain is sufficiently large.
Parallel repetition in projection games and a concentration bound
 In Proc. 40th STOC
, 2008
"... In a two player game, a referee asks two cooperating players (who are not allowed to communicate) questions sampled from some distribution and decides whether they win or not based on some predicate of the questions and their answers. The parallel repetition of the game is the game in which the refe ..."
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Cited by 42 (8 self)
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In a two player game, a referee asks two cooperating players (who are not allowed to communicate) questions sampled from some distribution and decides whether they win or not based on some predicate of the questions and their answers. The parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than (1 − ǫ), what’s the best they can do in the repeated game? We improve earlier results [Raz98, Hol07], which showed that the players cannot win all copies in the repeated game with probability better than (1 −ǫ 3) Ω(n/c) (here c is the length of the answers in the game), in the following ways: • We prove the bound (1 −ǫ 2) Ω(n) as long as the game is a “projection game”, the type of game most commonly used in hardness of approximation results. Our bound is independent of the answer length and has a better dependence on ǫ. By the recent work of Raz [Raz08], this bound is tight. A consequence of this bound is that the Unique Games Conjecture of Khot [Kho02] is equivalent to: Unique Games Conjecture There is an unbounded increasing function f: R + → R + such that for every ǫ> 0, there exists an alphabet size M(ǫ) for which it is NPhard to distinguish a Unique Game with alphabet size M in which a 1 −ǫ 2 fraction of the constraints can be satisfied from one in which a 1 − ǫf(1/ǫ) fraction of the constraints can be satisfied. • We prove a concentration bound for parallel repetition (of general games) showing that for any constant 0 < δ < ǫ, the probability that the players win a (1 −ǫ+δ) fraction of the games in the parallel repetition is at most exp � −Ω(δ 4 n/c) �. An application of this is in testing Bell Inequalities. Our result implies that the parallel repetition of the CHSH game can be used to get an experiment that has a very large classical versus quantum gap.
SPECTRAL ALGORITHMS FOR UNIQUE Games
"... We give a new algorithm for Unique Games which is based on purely spectral techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The appro ..."
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We give a new algorithm for Unique Games which is based on purely spectral techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The approximation guarantee depends only on the completeness of the game, and not on the alphabet size, while the running time depends on spectral properties of the LabelExtended graph associated with the instance of Unique Games. We further show that on input the integrality gap instance of Khot and Vishnoi, our algorithm runs in quasipolynomial time and decides that the instance if highly unsatisfiable. Notably, when run on this instance, the standard SDP relaxation of Unique Games fails. As a special case, we also rederive a polynomial time algorithm for Unique Games on expander constraint graphs. The main ingredient of our algorithm is a technique to effectively use the full spectrum of the underlying graph instead of just the second eigenvalue, which is of independent interest. The question of how to take advantage of the full spectrum of a graph in the design of algorithms has been often studied, but no significant progress was made prior to this work.
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.
Playing Random and Expanding Unique Games
"... We analyze the behavior of the SDP by Feige and Lovász [FL92] on random instances of unique games. We show that on random dregular graphs with permutations chosen at random, the value of the SDP is very small with probablility 1−e −Ω(d). Hence, the SDP provides a proof of unsatisfiability for rando ..."
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We analyze the behavior of the SDP by Feige and Lovász [FL92] on random instances of unique games. We show that on random dregular graphs with permutations chosen at random, the value of the SDP is very small with probablility 1−e −Ω(d). Hence, the SDP provides a proof of unsatisfiability for random unique games. We also give a spectral algorithm for recovering planted solutions. Given a random instance consistent with a given solution on 1 − ɛ fraction of the edges, our algorithm recovers a solution with value 1 − O(ɛ) with high probability at least 1 − e −Ω(d) over the inputs.
SPECIAL ISSUE: ANALYSIS OF BOOLEAN FUNCTIONS DimensionFree L2 Maximal Inequality for Spherical Means in the Hypercube
, 2013
"... Abstract: We establish the maximal inequality claimed in the title. In combinatorial terms this has the implication that for sufficiently small ε> 0, for all n, any marking of an ε fraction of the vertices of the ndimensional hypercube necessarily leaves a vertex x such that marked vertices are ..."
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Abstract: We establish the maximal inequality claimed in the title. In combinatorial terms this has the implication that for sufficiently small ε> 0, for all n, any marking of an ε fraction of the vertices of the ndimensional hypercube necessarily leaves a vertex x such that marked vertices are a minority of every sphere centered at x. ACM Classification: G.3 AMS Classification: 42B25 Key words and phrases: maximal inequality, Fourier analysis, boolean hypercube 1
Unique Games on the Hypercube
, 2014
"... In this paper, we investigate the validity of the Unique Games Conjecture when the constraint graph is the boolean hypercube. We construct an almost optimal integrality gap instance on the Hypercube for the GoemansWilliamson semidefinite program (SDP) for Max2LIN(Z2). We conjecture that adding tr ..."
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In this paper, we investigate the validity of the Unique Games Conjecture when the constraint graph is the boolean hypercube. We construct an almost optimal integrality gap instance on the Hypercube for the GoemansWilliamson semidefinite program (SDP) for Max2LIN(Z2). We conjecture that adding triangle inequalities to the SDP provides a polynomial time algorithm to solve Unique Games on the hypercube. 1