Parameterized Complexity of Path Set Packing

TL;DR

This paper investigates the parameterized complexity of Path Set Packing, establishing hardness results and providing fixed-parameter tractable algorithms based on various graph parameters, along with a 4-approximation algorithm.

Contribution

It answers open questions about the problem's complexity and introduces new FPT algorithms and approximation methods based on structural graph parameters.

Findings

W[1]-hardness with respect to vertex cover number

W[1]-hardness with respect to pathwidth plus max degree plus solution size

FPT algorithms parameterized by feedback vertex number plus max degree, and by treewidth plus max degree plus max path length

Abstract

In Path Set Packing, the input is an undirected graph $G$, a collection $\calp$ of simple paths in $G$, and a positive integer $k$. The problem is to decide whether there exist $k$ edge-disjoint paths in $\calp$. We study the parameterized complexity of Path Set Packing with respect to both natural and structural parameters. We show that the problem is $W[1]$-hard with respect to vertex cover number, and $W[1]$-hard respect to pathwidth plus maximum degree plus solution size. These results answer an open question raised in COCOON 2018. On the positive side, we present an FPT algorithm parameterized by feedback vertex number plus maximum degree, and present an FPT algorithm parameterized by treewidth plus maximum degree plus maximum length of a path in $\calp$. These positive results complement the hardness of Path Set Packing with respect to any subset of the parameters used in the FPT algorithms. We also give a $4$-approximation algorithm for maximum path set packing problem which runs in FPT time when parameterized by feedback edge number.