Component twin-width as a parameter for BINARY-CSP and its semiring generalisations

TL;DR

This paper introduces a generalized algebraic framework based on component twin-width for solving various binary constraint satisfaction problems efficiently, improving existing complexity bounds.

Contribution

It generalizes the twin-width parameter to edge-labelled graphs for arbitrary binary constraints and develops a unified FPT algorithm applicable to multiple problem variants.

Findings

Provides an FPT algorithm for problems with bounded component twin-width.

Achieves improved complexity bounds for several graph coloring and constraint satisfaction problems.

Demonstrates the framework's effectiveness on problems like H-coloring and its variants.

Abstract

We investigate the fine-grained and the parameterized complexity of several generalizations of binary constraint satisfaction problems (BINARY-CSPs), that subsume variants of graph colouring problems. Our starting point is the observation that several algorithmic approaches that resulted in complexity upper bounds for these problems, share a common structure. We thus explore an algebraic approach relying on semirings that unifies different generalizations of BINARY-CSPs (such as the counting, the list, and the weighted versions), and that facilitates a general algorithmic approach to efficiently solving them. The latter is inspired by the (component) twin-width parameter introduced by Bonnet et al., which we generalize via edge-labelled graphs in order to formulate it to arbitrary binary constraints. We consider input instances with bounded component twin-width, as well as constraint templates of bounded component twin-width, and obtain an FPT algorithm as well as an improved, exponential-time algorithm, for broad classes of binary constraints. We illustrate the advantages of this framework by instantiating our general algorithmic approach on several classes of problems (e.g., the $H$-coloring problem and its variants), and showing that it improves the best complexity upper bounds in the literature for several well-known problems.