Reconfiguration over tree decompositions

TL;DR

This paper studies the complexity of reconfiguring feasible solutions in vertex-subset problems on graphs with bounded treewidth, showing PSPACE-completeness generally but fixed-parameter tractability for MSOL-definable problems.

Contribution

It proves PSPACE-completeness of reconfiguration problems on bounded treewidth graphs and introduces a technique to handle MSOL-definable problems for fixed-parameter tractability.

Findings

Reconfiguration problems are PSPACE-complete on graphs of bounded treewidth.

MSOL-definable problems are fixed-parameter tractable when parameterized by solution steps and treewidth.

A new technique allows MSOL-based reconfiguration problem formulation despite cardinality constraints.

Abstract

A vertex-subset graph problem $Q$ defines which subsets of the vertices of an input graph are feasible solutions. The reconfiguration version of a vertex-subset problem $Q$ asks whether it is possible to transform one feasible solution for $Q$ into another in at most $\ell$ steps, where each step is a vertex addition or deletion, and each intermediate set is also a feasible solution for $Q$ of size bounded by $k$. Motivated by recent results establishing W[1]-hardness of the reconfiguration versions of most vertex-subset problems parameterized by $\ell$, we investigate the complexity of such problems restricted to graphs of bounded treewidth. We show that the reconfiguration versions of most vertex-subset problems remain PSPACE-complete on graphs of treewidth at most $t$ but are fixed-parameter tractable parameterized by $\ell + t$ for all vertex-subset problems definable in monadic second-order logic (MSOL). To prove the latter result, we introduce a technique which allows us to circumvent cardinality constraints and define reconfiguration problems in MSOL.