On $d$-stable locally checkable problems parameterized by mim-width

TL;DR

This paper investigates the complexity of locally checkable problems on graphs with bounded mim-width, introducing the concept of d-stability, and shows these problems are solvable in polynomial time given a suitable decomposition.

Contribution

It introduces d-stability for locally checkable problems, establishes their solvability in polynomial time on graphs of bounded mim-width, and links this framework with DN logic.

Findings

Problems are XP when parameterized by mim-width.

d-stability characterizes problems solvable efficiently on bounded mim-width graphs.

The framework unifies and extends previous complexity results for locally checkable problems.

Abstract

In this paper we continue the study of locally checkable problems under the framework introduced by Bonomo-Braberman and Gonzalez in 2020, by focusing on graphs of bounded mim-width. We study which restrictions on a locally checkable problem are necessary in order to be able to solve it efficiently on graphs of bounded mim-width. To this end, we introduce the concept of $d$-stability of a check function. The related locally checkable problems contain large classes of problems, among which we can mention, for example, LCVP problems. We give an algorithm showing that these problems are XP when parameterized by the mim-width of a given binary decomposition tree of the input graph, that is, that they can be solved in polynomial time given a binary decomposition tree of bounded mim-width. We explore the relation between $d$-stable locally checkable problems and the recently introduced DN logic (Bergougnoux, Dreier and Jaffke, 2022), and show that both frameworks model the same family of problems. We include a list of concrete examples of $d$-stable locally checkable problems whose complexity on graphs of bounded mim-width was open so far.