Testing the complexity of a valued CSP language

TL;DR

This paper investigates the complexity of testing algebraic conditions in valued constraint satisfaction problems (VCSPs), providing algorithms and lower bounds for determining tractability of language classes.

Contribution

It introduces an exponential algorithm for testing the tractability condition of finite-valued VCSP languages and establishes matching lower bounds under SETH.

Findings

Testing the tractability condition can be done in $O( oot{3}{3}^{|D|} imes poly)$ time.

A matching lower bound is proved assuming SETH, indicating the algorithm's near-optimality.

The results advance understanding of the complexity of classifying VCSP languages.

Abstract

A Valued Constraint Satisfaction Problem (VCSP) provides a common framework that can express a wide range of discrete optimization problems. A VCSP instance is given by a finite set of variables, a finite domain of labels, and an objective function to be minimized. This function is represented as a sum of terms where each term depends on a subset of the variables. To obtain different classes of optimization problems, one can restrict all terms to come from a fixed set $\Gamma$ of cost functions, called a language. Recent breakthrough results have established a complete complexity classification of such classes with respect to language $\Gamma$: if all cost functions in $\Gamma$ satisfy a certain algebraic condition then all $\Gamma$-instances can be solved in polynomial time, otherwise the problem is NP-hard. Unfortunately, testing this condition for a given language $\Gamma$ is known to be NP-hard. We thus study exponential algorithms for this meta-problem. We show that the tractability condition of a finite-valued language $\Gamma$ can be tested in $O(\sqrt[3]{3}^{\,|D|}\cdot poly(size(\Gamma)))$ time, where $D$ is the domain of $\Gamma$ and $poly(\cdot)$ is some fixed polynomial. We also obtain a matching lower bound under the Strong Exponential Time Hypothesis (SETH). More precisely, we prove that for any constant $\delta<1$ there is no $O(\sqrt[3]{3}^{\,\delta|D|})$ algorithm, assuming that SETH holds.