Parameterized Shortest Path Reconfiguration

TL;DR

This paper investigates the computational complexity of reconfiguring shortest s-t paths in graphs, revealing hardness results in general and fixed-parameter tractability in specific graph classes, advancing understanding of path reconfiguration problems.

Contribution

It demonstrates W[1]-hardness of the problem in bounded degeneracy graphs and establishes fixed-parameter tractability for certain graph parameters and classes.

Findings

W[1]-hardness when parameterized by path length and reconfiguration sequence length

Fixed-parameter tractability on nowhere-dense graph classes

FPT results for treedepth, cluster-deletion number, and modular-width

Abstract

An st-shortest path, or st-path for short, in a graph G is a shortest (induced) path from s to t in G. Two st-paths are said to be adjacent if they differ on exactly one vertex. A reconfiguration sequence between two st-paths P and Q is a sequence of adjacent st-paths starting from P and ending at Q. Deciding whether there exists a reconfiguration sequence between two given $st$-paths is known to be PSPACE-complete, even on restricted classes of graphs such as graphs of bounded bandwidth (hence pathwidth). On the positive side, and rather surprisingly, the problem is polynomial-time solvable on planar graphs. In this paper, we study the parameterized complexity of the Shortest Path Reconfiguration (SPR) problem. We show that SPR is W[1]-hard parameterized by k + \ell, even when restricted to graphs of bounded (constant) degeneracy; here k denotes the number of edges on an st-path, and \ell denotes the length of a reconfiguration sequence from P to Q. We complement our hardness result by establishing the fixed-parameter tractability of SPR parameterized by \ell and restricted to nowhere-dense classes of graphs. Additionally, we establish fixed-parameter tractability of SPR when parameterized by the treedepth, by the cluster-deletion number, or by the modular-width of the input graph.