The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

Caterina Viola,Stanislav Živný

doi:10.1145/3458041

Abstract

Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami (SODA’20) for promise (non-valued) CSPs (on finite domains).

Highlights

An instance of a Constraint satisfaction problems (CSPs) is defined by finitely many relations that must hold among finitely many given variables; the computational task is to decide whether it is possible to find an assignment of labels from a fixed set to the variables so that all the constraints are satisfied
There are many other problems in which some of the constraints may be violated at a cost or in which there are satisfying assignments which are preferable to others. These situations are captured by valued constraint satisfaction problems
We focus on the combined basic linear programming (BLP) and affine integer programming (AIP) relaxation introduced by Brakensiek and Guruswami [17]

Summary

Introduction

By extending the argument from [17], we establish a sufficient algebraic condition on the combined relaxation for the solvability of promise VCSPs in which the domain of the “weak cost functions” is possibly infinite (Theorem 4). The proof of this result draws on ideas introduced in [17] but requires a non-trivial amount of technical machinery to make it work in the infinite-domain valued setting. All details with more explanation and examples can be found in the full version of the paper [50]

Preliminaries

The Combined BLP and AIP Relaxation for PVCSPs

The Combined Relaxation with Sampling for PVCSPs