A characterization of strong approximation resistance

Khot, Subhash; Tulsiani, Madhur; Worah, Pratik

doi:10.1145/2591796.2591817

Cited by 19 publications

(26 citation statements)

References 45 publications

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“…The first optimal inapproximability results [23] by Håstad were about problems Max CSP(Γ), and they led to the study of a new hardness notion called approximation resistance (see, e.g. [3,24,28]). The approximability of Boolean CSPs has been thoroughly investigated (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

Towards a Characterization of Constant-Factor Approximable Min CSPs

Dalmau¹,

Krokhin²,

Manokaran³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe study the approximability of Minimum Constraint Satisfaction Problems (Min CSPs) with a fixed finite constraint language Γ on an arbitrary finite domain. The goal in such a problem is to minimize the number of unsatisfied constraints in a given instance of CSP(Γ). A recent result of Ene et al. says that, under the mild technical condition that Γ contains the equality relation, the basic LP relaxation is optimal for constant-factor approximation for Min CSP(Γ) unless the Unique Games Conjecture fails. Using the algebraic approach to the CSP, we introduce a new natural algebraic condition, stable probability distributions on symmetric polymorphisms of a constraint language, and show that the presence of such distributions on polymorphisms of each arity is necessary and sufficient for the finiteness of the integrality gap for the basic LP relaxation of Min CSP(Γ). We also show how stable distributions on symmetric polymorphisms can in principle be used to round solutions of the basic LP relaxation, and how, for several examples that cover all previously known cases, this leads to efficient constant-factor approximation algorithms for Min CSP(Γ). Finally, we show that the absence of another condition, which is implied by stable distributions, leads to NP-hardness of constant-factor approximation.

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Section: Introductionmentioning

confidence: 99%

Towards a Characterization of Constant-Factor Approximable Min CSPs

Dalmau¹,

Krokhin²,

Manokaran³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…These results establish NP-hardness of the discussed problems and if we settle for Unique Games hardness, Austrin and Mossel [4] gave very general conditions for approximation resistance that apply to a vast majority of all constraint predicates [3]. Going even further, also in the case of Unique Games hardness, Khot et al [16], gave necessary and sufficient conditions for "strong approximation resistance," a slightly stronger notion and thus we have, in broad terms, a fairly good understanding of the complexity of approximating CSPs. There are two special cases where the picture is less clear and more information is needed.…”

Section: Introductionmentioning

confidence: 53%

“…We note THEORY OF COMPUTING, Volume 13 (10), 2017, pp. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] that an inapproximability of homogeneous linear equations, albeit with a stronger guarantee on the structure of the system, forms the core of the inapproximability of Min-Bisection [14].…”

Section: Introductionmentioning

confidence: 99%

Untitled

Håstad¹,

Manokaran²

2017

Theory of Comput.

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“…After proving our main result, we switch gears and analyze balanced LTFs which are approximable. In section 5 we give a description based on the analysis of Khot, Tulsiani, and Worah [16] for the space of approximation algorithms we should search over. In section 6 we use this description to give a simpler approximation algorithm for the monarchy predicate.…”

Section: Outlinementioning

confidence: 99%

On the approximation resistance of balanced linear threshold functions

Potechin

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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In this paper, we show that there exists a balanced linear threshold function (LTF) which is unique games hard to approximate, refuting a conjecture of Austrin, Benabbas, and Magen. We also show that the almost monarchy predicate on k variables is approximable for sufficiently large k. . arXiv:1807.04421v2 [cs.CC] 13 Dec 2018 2 PreliminariesIn this section, we give some preliminary definitions. In particular, we define what we mean by predicates, constraints, and approximation resistance and recall the unique games problem and the unique games conjecture.Definition 2.1. A boolean predicate P of arity k is a function P :Definition 2.3. Let P be a boolean predicate. We say that a boolean constraint C : {−1, +1} n → {−1, +1} has form P if there exists a map φ : [1, k] → [1, n] and signs (z 1 , . . . , z k ) ∈ {−1, +1} k such that C(x 1 , . . . , x n ) = P (z 1 x φ(1) , . . . , z k x φ(k) ) Definition 2.4. We say that a boolean predicate P is approximable if there exists an > 0 such that there is a polynomial time algorithm which takes a CSP with m constraints of form P as input and can distinguish between the following two cases:1. At least (1 − )m of the constraints can be satisfied.

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A characterization of strong approximation resistance

Cited by 19 publications

References 45 publications

Towards a Characterization of Constant-Factor Approximable Min CSPs

Towards a Characterization of Constant-Factor Approximable Min CSPs

Untitled

On the approximation resistance of balanced linear threshold functions

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