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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.
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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Cited by 17 (1 self)
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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.
Merging Techniques for Combinatorial Optimization: Spectral Graph Theory and Semidefinite Programming
, 2009
"... In this thesis, we study three problems related to expanders, whose analysis involves understanding the intimate connection between expanders, spectra and semidefinite programming. Our first result is related to Khot’s Unique Games conjecture (UGC) [Kho], whose validity is one of the most central op ..."
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In this thesis, we study three problems related to expanders, whose analysis involves understanding the intimate connection between expanders, spectra and semidefinite programming. Our first result is related to Khot’s Unique Games conjecture (UGC) [Kho], whose validity is one of the most central open problems in computational complexity theory. We show that UGC is false on expander graphs. This result, in particular, rules out a natural way of proving hardness of approximation for SPARSEST CUT. Our second result is in the area of graph sparsification. We say that a graph H is a sparsifier for a graph G if the respective graph Laplacians of the two graphs satisfy x T LHx ≈ x T LGx for all vectors x. Given a union of two graphs G + W, we show how to choose a sparse subgraph W ′ ⊆ W so that G + W ′ is a good sparsifier for G + W. We apply the result to optimizing the algebraic connectivity of a graph by adding very few edges. We also show how to use this result in order to create optimal ultrasparsifiers for every graph, which can be used as good graphtheoretic preconditioners for symmetric, positive semidefinite, diagonally dominant linear systems. Lastly, we study the integrality gap of the well known Sparsest Cut semidefinite program. We present a simple construction and analysis of an Ω(log log N) integrality gap
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