Two new homomorphism dualities and lattice operations

Carvalho, Catarina; Dalmau, Víctor; Krokhin, Andrei

doi:10.1093/logcom/exq030

Cited by 9 publications

(12 citation statements)

References 31 publications

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“…The first of these examples covers all problems Min CSP(Γ) that were previously known to belong to APX. Two classes of CSPs were introduced and studied in [10], one is a subclass of the other. We need two notions to define these classes.…”

Section: Algorithmsmentioning

confidence: 99%

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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: Algorithmsmentioning

confidence: 99%

“…Constraint languages k -IHBS (defined in Section 1) belong to the larger, but not to the smaller class (see Example 1). See [10] for other specific examples of CSPs contained in these classes. For the smaller class, Min CSP(Γ) was shown to belong to APX in [32].…”

Section: Algorithmsmentioning

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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“…We first describe the characterization of lattice CSPs we need. Carvalho, Dalmau, and Krokhin [5] have observed that if CSP(Γ) has lattice polymorphisms then it is preserved by what they call an absorptive block-symmetric operation. This is an operation f which takes as input tuples (of any…”

Section: A Lattice Csps: Better Quantitative Dependence Onmentioning

confidence: 99%

Linear programming, width-1 CSPs, and robust satisfaction

Kun

O’Donnell

Tamaki

et al. 2012

Proceedings of the 3rd Innovations in Theoretical Computer Science Conference

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“…Another family of problems CSP(Γ) with bounded pathwidth duality was shown to admit robust algorithms with polynomial loss in [23], where the parameter k depends on the pathwidth duality bound (and appears in the algebraic description of this family). This family includes languages not having an NU polymorphism of any arity -see [13,14]. It is unclear how far connections between the two directions go, but consistency notions seem to be the common theme.…”

Section: Resultsmentioning

confidence: 99%

Robust algorithms with polynomial loss for near-unanimity CSPs

Dalmau

Kozik

Krokhin

et al. 2017

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

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An instance of the Constraint Satisfaction Problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a fixed domain to the variables so that all constraints are satisfied. In the optimization version, the goal is to maximize the number of satisfied constraints. An approximation algorithm for CSP is called robust if it outputs an assignment satisfying a (1 − g(ε))-fraction of constraints on any (1 − ε)-satisfiable instance, where the loss function g is such that g(ε) → 0 as ε → 0.We study how the robust approximability of CSPs depends on the set of constraint relations allowed in instances, the so-called constraint language. All constraint languages admitting a robust polynomial-time algorithm (with some g) have been characterised by Barto and Kozik, with the general bound on the loss g being doubly exponential, specifically g(ε) = O((log log(1/ε))/ log(1/ε)). It is natural to ask when a better loss can be achieved: in particular, polynomial loss g(ε) = O(ε 1/k ) for some constant k. In this paper, we consider CSPs with a constraint language having a nearunanimity polymorphism. We give two randomized robust algorithms with polynomial loss for such CSPs: one works for any near-unanimity polymorphism and the parameter k in the loss depends on the size of the domain and the arity of the relations in Γ, while the other works for a special ternary near-unanimity operation called dual discriminator with k = 2 for any domain size. In the latter case, the CSP is a common generalisation of Unique Games with a fixed domain and 2-Sat. In the former case, we use the algebraic approach to the CSP. Both cases use the standard semidefinite programming relaxation for CSP.

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Two new homomorphism dualities and lattice operations

Cited by 9 publications

References 31 publications

Towards a Characterization of Constant-Factor Approximable Min CSPs

Towards a Characterization of Constant-Factor Approximable Min CSPs

Linear programming, width-1 CSPs, and robust satisfaction

Robust algorithms with polynomial loss for near-unanimity CSPs

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