Backdoors to Tractable Valued CSP

TL;DR

This paper introduces the concept of strong backdoors for Valued CSPs, enabling efficient solutions for instances close to tractable classes by measuring the minimal backdoor size, and extends fixed-parameter tractability results to scattered classes.

Contribution

It generalizes the notion of strong backdoors to VCSPs and proves fixed-parameter tractability for instances near tractable classes, including scattered classes with components from different tractable classes.

Findings

VCSP is fixed-parameter tractable when parameterized by backdoor size.

Extension of fixed-parameter tractability to scattered classes of VCSP.

Backdoors enable solving instances close to tractable VCSP classes.

Abstract

We extend the notion of a strong backdoor from the CSP setting to the Valued CSP setting (VCSP, for short). This provides a means for augmenting a class of tractable VCSP instances to instances that are outside the class but of small distance to the class, where the distance is measured in terms of the size of a smallest backdoor. We establish that VCSP is fixed-parameter tractable when parameterized by the size of a smallest backdoor into every tractable class of VCSP instances characterized by a (possibly infinite) tractable valued constraint language of finite arity and finite domain. We further extend this fixed-parameter tractability result to so-called "scattered classes" of VCSP instances where each connected component may belong to a different tractable class.