Towards a Characterization of Constant-Factor Approximable Min CSPs

Dalmau, V.; Krokhin, A.; Manokaran, R.

Abstract

We 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.

Citation

Dalmau, V., Krokhin, A., & Manokaran, R. (2015). Towards a Characterization of Constant-Factor Approximable Min CSPs. In P. Indyk (Ed.), Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms : San Diego, California, USA, January 4-6, 2015 (847-857). Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611973730.58

• Online Publication Date: Dec 22, 2014
• Publication Date: Jan 1, 2015
• Deposit Date: Dec 15, 2015
• Publicly Available Date: Mar 23, 2016
• Publisher: Society for Industrial and Applied Mathematics
• Pages: 847-857
• Series Title: Proceedings
• Book Title: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms : San Diego, California, USA, January 4-6, 2015.
• ISBN: 9781611973747
• DOI: https://doi.org/10.1137/1.9781611973730.58
• Public URL: https://durham-repository.worktribe.com/output/1668035
• Contract Date: Sep 16, 2014