Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited

TL;DR

This paper establishes new fixed-parameter tractability results for combinatorial reconfiguration problems using structural graph parameters like neighborhood diversity and treedepth, especially when the number of steps is not part of the parameter.

Contribution

It provides the first algorithmic meta-theorems for reconfiguration problems with fixed parameters excluding the number of steps, using parameters like neighborhood diversity and treedepth.

Findings

Reconfiguration is FPT parameterized by neighborhood diversity.

Reconfiguration is FPT parameterized by treedepth + set size.

Reconfiguration is PSPACE-complete on forests of depth 3.

Abstract

Given a graph and two vertex sets satisfying a certain feasibility condition, a reconfiguration problem asks whether we can reach one vertex set from the other by repeating prescribed modification steps while maintaining feasibility. In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic meta-theorem for reconfiguration problems that says if the feasibility can be expressed in monadic second-order logic (MSO), then the problem is fixed-parameter tractable parameterized by $\textrm{treewidth} + \ell$, where $\ell$ is the number of steps allowed to reach the target set. On the other hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if $\ell$ is not part of the parameter, then the problem is PSPACE-complete even on graphs of bounded bandwidth. In this paper, we present the first algorithmic meta-theorems for the case where $\ell$ is not part of the parameter, using some structural graph parameters incomparable with bandwidth. We show that if the feasibility is defined in MSO, then the reconfiguration problem under the so-called token jumping rule is fixed-parameter tractable parameterized by neighborhood diversity. We also show that the problem is fixed-parameter tractable parameterized by $\textrm{treedepth} + k$, where $k$ is the size of sets being transformed. We finally complement the positive result for treedepth by showing that the problem is PSPACE-complete on forests of depth $3$.