Finding Shortest Paths between Graph Colourings

TL;DR

This paper thoroughly analyzes the parameterized complexity of the $k$-colouring reconfiguration problem, providing polynomial-time solutions for $k=3$ and fixed-parameter tractability results for $k \\geq 4$, with kernelization limitations.

Contribution

It resolves an open problem by showing polynomial-time solvability for $k=3$ and establishes fixed-parameter tractability with kernelization bounds for all $k \\geq 4$.

Findings

Polynomial-time solvability for $k=3$

Fixed-parameter tractability for $k \\geq 4$

No polynomial kernel unless the polynomial hierarchy collapses

Abstract

The $k$-colouring reconfiguration problem asks whether, for a given graph $G$, two proper $k$-colourings $\alpha$ and $\beta$ of $G$, and a positive integer $\ell$, there exists a sequence of at most $\ell+1$ proper $k$-colourings of $G$ which starts with $\alpha$ and ends with $\beta$ and where successive colourings in the sequence differ on exactly one vertex of $G$. We give a complete picture of the parameterized complexity of the $k$-colouring reconfiguration problem for each fixed $k$ when parameterized by $\ell$. First we show that the $k$-colouring reconfiguration problem is polynomial-time solvable for $k=3$, settling an open problem of Cereceda, van den Heuvel and Johnson. Then, for all $k \geq 4$, we show that the $k$-colouring reconfiguration problem, when parameterized by $\ell$, is fixed-parameter tractable (addressing a question of Mouawad, Nishimura, Raman, Simjour and Suzuki) but that it has no polynomial kernel unless the polynomial hierarchy collapses.