The Power of Cut-Based Parameters for Computing Edge Disjoint Paths

TL;DR

This paper explores how structural graph parameters, especially treecut width and feedback edge set size, influence the complexity of the Edge Disjoint Paths problem, providing new algorithms and kernelization results.

Contribution

It introduces a polynomial-time algorithm for EDP on graphs with bounded treecut width and establishes a polynomial kernel based on feedback edge set size, advancing understanding of EDP's complexity.

Findings

Polynomial-time algorithm for EDP on graphs with bounded treecut width

EDP parameterized by treecut width is unlikely to be fixed-parameter tractable

Polynomial kernel for EDP based on minimum feedback edge set size

Abstract

This paper revisits the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph $G$ and a set of terminal pairs $P$ and asks whether $G$ contains a set of pairwise edge-disjoint paths connecting every terminal pair in $P$. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in $P$ in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size $3$. We present three results which use edge-separator based parameters to chart new islands of tractability in the complexity landscape of EDP. Our first and main result utilizes the fairly recent structural parameter treecut width (a parameter with fundamental ties to graph immersions and graph cuts): we obtain a polynomial-time algorithm for EDP on every graph class of bounded treecut width. Our second result shows that EDP parameterized by treecut width is unlikely to be fixed-parameter tractable. Our final, third result is a polynomial kernel for EDP parameterized by the size of a minimum feedback edge set in the graph.