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46
Optimal inapproximability results for MAXCUT and other 2variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 238 (32 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games
On the Hardness of Approximating Multicut and SparsestCut
 In Proceedings of the 20th Annual IEEE Conference on Computational Complexity
, 2005
"... We show that the MULTICUT, SPARSESTCUT, and MIN2CNF ≡ DELETION problems are NPhard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1. ..."
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Cited by 102 (5 self)
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We show that the MULTICUT, SPARSESTCUT, and MIN2CNF ≡ DELETION problems are NPhard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1.
Conditional hardness for approximate coloring
 In STOC 2006
, 2006
"... We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For ..."
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Cited by 46 (13 self)
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We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For q = 3, we base our hardness result on a certain ‘⊲< shaped ’ variant of his conjecture. We also prove that the problem ALMOST3COLORINGε is hard for any constant ε> 0, assuming Khot’s Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3color all but a ε fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least ε. Our result is based on bounding various generalized noisestability quantities using the invariance principle of Mossel et al [MOO’05].
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.
Integrality gaps for strong SDP relaxations of unique games
"... Abstract — With the work of Khot and Vishnoi [18] as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner pro ..."
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Cited by 42 (8 self)
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Abstract — With the work of Khot and Vishnoi [18] as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to exp(Ω(log log n) 1/4) vectors. For a stronger relaxation obtained from the basic semidefinite program by R rounds of Sherali–Adams liftandproject, we prove a Unique Games integrality gap for R = Ω(log log n) 1/4. By composing these SDP gaps with UGChardness reductions, the above results imply corresponding integrality gaps for every problem for which a UGCbased hardness is known. Consequently, this work implies that including any valid constraints on up to exp(Ω(log log n) 1/4) vectors to natural semidefinite program, does not improve the approximation ratio for any problem in the following classes: constraint satisfaction problems, ordering constraint satisfaction problems and metric labeling problems over constantsize metrics. We obtain similar SDP integrality gaps for Balanced Separator, building on [11]. We also exhibit, for explicit constants γ, δ> 0, an npoint negativetype metric which requires distortion Ω(log log n) γ to embed into ℓ1, although all its subsets of size exp(Ω(log log n) δ) embed isometrically into ℓ1. Keywordssemidefinite programming, approximation algorithms, unique games conjecture, hardness of approximation, SDP hierarchies, Sherali–Adams hierarchy, integrality gap construction 1.
Unique games with entangled provers are easy
 In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
, 2008
"... We consider oneround games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are ‘unique ’ constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only a ..."
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Cited by 29 (8 self)
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We consider oneround games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are ‘unique ’ constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only algorithm known previously was for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things, our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false. Our proof is based on a novel ‘quantum rounding technique’, showing how to take a solution to an SDP and transform it to a strategy for entangled provers. Using our approximation by a semidefinite program we also show a parallel repetition theorem for unique entangled games. 1
Approximating unique games
 In Proc. SODA’06
, 2006
"... The Unique Games problem is the following: we are given a graph G = (V, E), with each edge e = (u, v) having a weight we and a permutation πuv on [k]. The objective is to find a labeling of each vertex u with a label fu ∈ [k] to minimize the weight of unsatisfied edges—where an edge (u, v) is satisf ..."
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Cited by 24 (1 self)
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The Unique Games problem is the following: we are given a graph G = (V, E), with each edge e = (u, v) having a weight we and a permutation πuv on [k]. The objective is to find a labeling of each vertex u with a label fu ∈ [k] to minimize the weight of unsatisfied edges—where an edge (u, v) is satisfied if fv = πuv(fu). The Unique Games Conjecture of Khot [8] essentially says that for each ε> 0, there is a k such that it is NPhard to distinguish instances of Unique games with (1−ε) satisfiable edges from those with only ε satisfiable edges. Several hardness results have recently been proved based on this assumption, including optimal ones for MaxCut, VertexCover and other problems, making it an important challenge to prove or refute the conjecture. In this paper, we give an O(log n)approximation algorithm for the problem of minimizing the number of unsatisfied edges in any Unique game. Previous results of Khot [8] and Trevisan [12] imply that if the optimal solution has OPT = εm unsatisfied edges, semidefinite relaxations of the problem could give labelings with min{k2ε1/5, (ε log n) 1/2}m unsatisfied edges. In this paper we show how to round a LP relaxation to get an O(log n)approximation to the problem; i.e., to find a labeling with only O(εm log n) = O(OPT log n) unsatisfied edges. 1
Faster generation of random spanning trees
"... Abstract — In this paper, we set forth a new algorithm for generating approximately uniformly random spanning trees in undirected graphs. We show how to sample from a distribution that is within a multiplicative (1 + δ) of uniform in expected time e O(m √ n log 1/δ). This improves the sparse graph c ..."
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Cited by 20 (2 self)
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Abstract — In this paper, we set forth a new algorithm for generating approximately uniformly random spanning trees in undirected graphs. We show how to sample from a distribution that is within a multiplicative (1 + δ) of uniform in expected time e O(m √ n log 1/δ). This improves the sparse graph case of the best previously known worstcase bound of O(min{mn, n 2.376}), which has stood for twenty years. To achieve this goal, we exploit the connection between random walks on graphs and electrical networks, and we use this to introduce a new approach to the problem that integrates discrete random walkbased techniques with continuous linear algebraic methods. We believe that our use of electrical networks and sparse linear system solvers in conjunction with random walks and combinatorial partitioning techniques is a useful paradigm that will find further applications in algorithmic graph theory. Keywordsspanning trees; random walks on graphs; electrical flows; 1.
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.