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16
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 216 (30 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 96 (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.
Nearoptimal algorithms for Unique Games
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... Unique games are constraint satisfaction problems that can be viewed as a generalization of MaxCut to a larger domain size. The Unique Games Conjecture states that it is hard to distinguish between instances of unique games where almost all constraints are satisfiable and those where almost none ar ..."
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Cited by 44 (8 self)
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Unique games are constraint satisfaction problems that can be viewed as a generalization of MaxCut to a larger domain size. The Unique Games Conjecture states that it is hard to distinguish between instances of unique games where almost all constraints are satisfiable and those where almost none are satisfiable. It has been shown to imply a number of inapproximability results for fundamental problems that seem difficult to obtain by more standard complexity assumptions. Thus, proving or refuting this conjecture is an important goal. We present significantly improved approximation algorithms for unique games. For instances with domain size k where the optimal solution satisfies 1 − ε fraction of all constraints, our algorithms satisfy roughly k −ε/(2−ε) and 1 − O ( √ ε log k) fraction of all constraints. Our algorithms are based on rounding a natural semidefinite programming relaxation for the problem and their performance almost matches the integrality gap of this relaxation. Our results are near optimal if the Unique Games Conjecture is true, i.e. any improvement (beyond low order terms) would refute the conjecture. 1
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.
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 23 (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
Every 2CSP Allows Nontrivial Approximation
"... We use semidefinite programming to prove that any constraint satisfaction problem in two variables over any domain allows an efficient approximation algorithm that does better than picking a random assignment. Specifically we consider the case when each variable can take values in [d] and that each ..."
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Cited by 18 (3 self)
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We use semidefinite programming to prove that any constraint satisfaction problem in two variables over any domain allows an efficient approximation algorithm that does better than picking a random assignment. Specifically we consider the case when each variable can take values in [d] and that each constraint rejects t out of the d2 possible input pairs. Then, for some universal constant c, wecan,in probabilistic polynomial time, find an assignment whose objective value is, in expectation, within a factor 1 − t d2 + ct d4 log d of optimal, improving on the trivial bound of 1 − t/d².
On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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Cited by 15 (1 self)
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
On the hardness of approximating MaxSatisfy
 Electronic Colloquium on Computational Complexity (ECCC
, 2004
"... MaxSatisfy is the problem of finding an assignment that satisfies the maximum number of equations in a system of linear equations over Q. We prove that unless NP⊂BPP MaxSatisfy cannot be efficiently approximated within an approximation ratio of 1/n 1−ɛ, if we consider systems of n linear equations ..."
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Cited by 9 (1 self)
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MaxSatisfy is the problem of finding an assignment that satisfies the maximum number of equations in a system of linear equations over Q. We prove that unless NP⊂BPP MaxSatisfy cannot be efficiently approximated within an approximation ratio of 1/n 1−ɛ, if we consider systems of n linear equations with at most n variables and ɛ> 0 is an arbitrarily small constant. Previously, it was known that the problem is NPhard to approximate within a ratio of 1/n α, but 0 < α < 1 was some specific constant that could not be taken to be arbitrarily close to 1.
Locally Expanding Hypergraphs and the Unique Games Conjecture
, 2008
"... We examine the hardness of approximating constraint satisfaction problems with kvariable constraints, known as kCSP’s. We are specifically interested in kCSP’s whose constraints are unique, which means that for any assignment to any k − 1 of the variables, there is a unique assignment to the last ..."
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We examine the hardness of approximating constraint satisfaction problems with kvariable constraints, known as kCSP’s. We are specifically interested in kCSP’s whose constraints are unique, which means that for any assignment to any k − 1 of the variables, there is a unique assignment to the last variable satisfying the constraint. One fundamental example of these CSP’s is EkLinp, the problem of satisfying as many equations as possible from an overdetermined system of linear equations modulo a prime p, where each equation contains exactly k variables. The central question in much of the recent work on inapproximability has been the Unique Games Conjecture, which posits a very strong hardness of approximation for 2CSP’s with unique constraints, such as E2Linp. Many strong inapproximability results have been proven assuming that it is true, including a recent result of Raghavendra (“Optimal algorithms and inapproximability results for every CSP? ” in STOC. ACM, 2008.) giving an approximation algorithm for every CSP, whose performance is essentially optimal if the Unique Games Conjecture is true. To date, however, not much progress has been made on resolving the conjecture. In this paper, we give a reduction from unique 3CSP’s to unique 2CSP’s which is sometimes approximationpreserving, depending on the combinatorial structure of the underlying hypergraph of the 3CSP. The underlying hypergraph of a 3CSP is the hypergraph in which each vertex represents a variable, and every hyperedge represents a constraint. Every constraint c yields a hyperedge between the three variables involved in c. The reduction only works when the underlying hypergraph of the 3CSP satisfies a (hypergraph) expansion property, which we call local expansion. We prove that the Unique Games Conjecture is equivalent to a hardness result for unique 3CSP’s whose underlying hypergraphs are local expanders. We give a precise formulation of the desired hardness result as a conjecture, which we call the Expanding Unique 3CSP’s Conjecture. We also give a restricted, but still equivalent, conjecture that E3Linp is
Lower bounds for Grothendieck problems
"... Given a graph G = (V;E), consider the following problem: The input is a function A: E! R, and the goal is to maximize P (u;v)2E A(u; v)f(u)f(v) over all functions f: V! f¡1; 1g. This problem was formalized by Alon, Makarychev, Makarychev and Naor [AMMN05]; it is a weighted version of the 2cluster \ ..."
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Given a graph G = (V;E), consider the following problem: The input is a function A: E! R, and the goal is to maximize P (u;v)2E A(u; v)f(u)f(v) over all functions f: V! f¡1; 1g. This problem was formalized by Alon, Makarychev, Makarychev and Naor [AMMN05]; it is a weighted version of the 2cluster \correlation clustering " problem introduced by Bansal, Blum, and Chawla [BBC02]. The most natural two instances are G = KN (the complete graph) and G = KN;N (the complete bipartite graph); the former includes MaxCutGain as a special case, the latter includes a form of CutNorm as a special case. We study lower bounds for these problems  mainly computational lower bounds based on the Unique Games Conjecture (UGC) [Kho02], but also semide¯nite programming (SDP) integrality gaps. Our lower bounds in the general KN case hold even for MaxCutGain: we give both a semide¯nite programming gap and a UGChardness result at versus ( = log(1=)) for any constant > 0. This matches the SDP algorithm of Charikar and Wirth [CW04]. Our lower bounds in the bipartite KN;N hold even in the case of CutNorm for zerosum matrices; we show how to translate the best known lower bound on Grothendieck's constant (due to Reeds [Ree93]) into a UGChardness result. This complements the work of Alon and Naor [AN04] who give an SDP approximation algorithm translating the best known upper bound on Grothendieck's constant (due to Krivine [Kri77]). 0 1