On LP-based Approximability for Strict CSPs

Kumar, Amit; Manokaran, Rajsekar; Tulsiani, Madhur; Vishnoi, Nisheeth K.

doi:10.1137/1.9781611973082.121

Cited by 22 publications

(43 citation statements)

References 19 publications

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“…The best currently known approximation algorithms for MinUnCut and Min 2CNF Deletion have approximation ratio O( log n) [1], and it follows from [26] that neither problem belongs to APX unless the UG conjecture is false. The UGC is known to imply the optimality of the basic LP relaxation for any Min CSP(Γ) such that Γ contains the equality relation (in fact, even for the more general Valued CSP) [18], extending the line of similar results for natural LP and SDP relaxations for various optimization CSPs [30,34,37]. An approximation algorithm for any Min CSP(Γ) is also given in [18] (that was claimed to match the LP integrality gap), but its analysis was later found to be faulty [41].…”

Section: Introductionmentioning

confidence: 89%

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: 89%

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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“…Recently, Bansal and Khot [3] and Kumar et al [11] independently showed that if the Unique Games Conjecture holds, PD w j C j is in fact inapproximable within a factor of 2 − ε for any ε > 0, unless P = NP. In [3], this result is obtained by combining our inapproximability construction with a new and stronger inapproximability result for the minimum vertex cover problem on runiform hypergraphs.…”

Section: Introductionmentioning

confidence: 99%

“…In [3], this result is obtained by combining our inapproximability construction with a new and stronger inapproximability result for the minimum vertex cover problem on runiform hypergraphs. In [11], this result is obtained by combining our integrality gap construction with an integrality-gap-based inapproximability result for strict constraint satisfaction problems.…”

Section: Introductionmentioning

confidence: 99%

Minimizing the sum of weighted completion times in a concurrent open shop

Mastrolilli

Queyranne

Schulz

et al. 2010

Operations Research Letters

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Hardness of approximation a b s t r a c tWe study minimizing the sum of weighted completion times in a concurrent open shop. We give a primal-dual 2-approximation algorithm for this problem. We also show that several natural linear programming relaxations for this problem have an integrality gap of 2. Finally, we show that this problem is inapproximable within a factor strictly less than 6/5 if P = NP, or strictly less than 4/3 if the Unique Games Conjecture also holds.

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“…The best currently known approximation algorithms for MinUnCut and Min 2CNF Deletion have approximation ratio O( log n) [29], and it follows from [32] that neither problem belongs to APX unless the UGC is false. The UGC is known to imply the optimality of the basic LP relaxation for any VCSP(Γ) such that Γ contains the (characteristic function of the) equality relation [20], extending the line of similar results for natural LP and SDP relaxations for various optimization CSPs [34,35,17]. An approximation algorithm for any VCSP(Γ) was also given in the 2013 conference version of [20] (that was claimed to match the LP integrality gap), but its analysis was later found to be faulty and this part was retracted in the 2015 update of [20].…”

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confidence: 99%

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confidence: 99%

Towards a characterization of constant-factor approximable finite-valued CSPs

Dalmau

Krokhin

Manokaran

2018

Journal of Computer and System Sciences

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In this paper we study the approximability of (Finite-)Valued Constraint Satisfaction Problems (VCSPs) with a fixed finite constraint language Γ consisting of finitary functions on a fixed finite domain. An instance of VCSP is given by a finite set of variables and a sum of functions belonging to Γ and depending on a subset of the variables. Each function takes values in [0, 1] specifying costs of assignments of labels to its variables, and the goal is to find an assignment of labels to the variables that minimizes the sum. A recent result of Ene et al. says that, under the mild technical condition that Γ contains the function corresponding to the equality relation, the basic LP relaxation is optimal for constant-factor approximation for VCSP(Γ) unless the Unique Games Conjecture fails. Using the algebraic approach to the CSP, we give new natural algebraic conditions for the finiteness of the integrality gap for the basic LP relaxation of VCSP(Γ). We also show how these algebraic conditions can in principle be used to round solutions of the basic LP relaxation, and how this leads to efficient constant-factor approximation algorithms for several examples that cover all previously known cases that are NP-hard to solve to optimality but admit constant-factor approximation. Finally, we show that the absence of another algebraic condition leads to NP-hardness of constant-factor approximation. Thus, our results strongly indicate where the boundary of constant-factor approximability for VCSPs lies. ✩ This article is an extended version of a paper published in the proceedings of SODA'15. (Rajsekar Manokaran) 21, 32]). Much work around the Unique Games Conjecture (UGC) directly concerns CSPs [21]. This conjecture states that, for any ǫ > 0, there is a large enough number k = k(ǫ) such that it NP-hard to tell ǫ-satisfiable from (1 − ǫ)-satisfiable instances of CSP(Γ k ), where Γ k consists of all graphs of bijections on a k-element set. Many approximation algorithms for classical optimization problems have been shown optimal assuming the UGC [21,32]. Raghavendra proved [17] that one SDP-based algorithm provides optimal approximation for all problems GCSP(Γ) assuming the UGC. In this paper, we investigate problems VCSP(Γ) and Min CSP(Γ) on an arbitrary finite domain that belong to APX, i.e. admit a (polynomial-time) constant-factor approximation algorithm, proving some results that strongly indicate where the boundary of this property lies.Related Work. Note that each problem Max CSP(Γ) trivially admits a constantfactor approximation algorithm because a random assignment of values to the variables is guaranteed to satisfy a constant fraction of constraints; this can be derandomized by the standard method of conditional probabilities. The same also holds for GCSP. Clearly, for Min CSP(Γ) to admit a constant-factor approximation algorithm, CSP(Γ) must be polynomial-time solvable.The approximability of problems VCSP(Γ) has been studied, mostly for Min CSPs in the Boolean case (i.e., with domain {0, 1}, such CSPs are sometimes ...

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On LP-based Approximability for Strict CSPs

Cited by 22 publications

References 19 publications

Towards a Characterization of Constant-Factor Approximable Min CSPs

Towards a Characterization of Constant-Factor Approximable Min CSPs

Minimizing the sum of weighted completion times in a concurrent open shop

Towards a characterization of constant-factor approximable finite-valued CSPs

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