A Definability Dichotomy for Finite Valued CSPs

TL;DR

This paper refines the complexity classification of finite valued CSPs by establishing a definability dichotomy, showing that polynomial-time solvable cases are definable in fixed-point logic with counting, while NP-hard cases are not.

Contribution

It introduces a definability-based dichotomy for finite valued CSPs, providing a logical characterization of tractable and intractable cases without relying on complexity assumptions.

Findings

Polynomial-time solvable cases are definable in fixed-point logic with counting.

NP-hard cases are not definable even in infinitary logic with counting.

The dichotomy is unconditional and based on logical definability.

Abstract

Finite valued constraint satisfaction problems are a formalism for describing many natural optimization problems, where constraints on the values that variables can take come with rational weights and the aim is to find an assignment of minimal cost. Thapper and Zivny have recently established a complexity dichotomy for finite valued constraint languages. They show that each such language either gives rise to a polynomial-time solvable optimization problem, or to an NP-hard one, and establish a criterion to distinguish the two cases. We refine the dichotomy by showing that all optimization problems in the first class are definable in fixed-point language with counting, while all languages in the second class are not definable, even in infinitary logic with counting. Our definability dichotomy is not conditional on any complexity-theoretic assumption.