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

• Published in: Journal of computer and system sciences Vol. 97; pp. 14 - 27
• Main Authors: Dalmau, Víctor, Krokhin, Andrei, Manokaran, Rajsekar
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.11.2018
• Subjects: Approximation algorithms; Constraint satisfaction problem; Universal algebra; Constraint satisfaction problem; Universal algebra; Approximation algorithms
• ISSN: 0022-0000; 1090-2724
• DOI: 10.1016/j.jcss.2018.03.003

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. Ene et al. have shown that, under a mild technical condition, 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) and show 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 indicate where the boundary of constant-factor approximability for VCSPs lies. •New natural algebraic conditions for the finiteness of the integrality gap for the basic LP relaxation of VCSP are given.•Efficient constant-factor approximation algorithms using the algebraic conditions are given.•Algebraic conditions for NP-hardness of constant-factor approximation are given.