The Parameterized Complexity Landscape of the Unsplittable Flow Problem

TL;DR

This paper provides a comprehensive parameterized complexity classification for the unsplittable flow problem on general graphs, identifying tractable cases and hardness results based on structural parameters.

Contribution

It introduces a full classification of the problem's complexity with respect to key structural parameters, including algorithms and hardness results.

Findings

Fixed-parameter algorithm for combined parameters

XP-tractability and W[1]-hardness results for individual parameters

Remaining cases are paraNP-hard

Abstract

We study the well-established problem of finding an optimal routing of unsplittable flows in a graph. While by now there is an extensive body of work targeting the problem on graph classes such as paths and trees, we aim at using the parameterized paradigm to identify its boundaries of tractability on general graphs. We develop novel algorithms and lower bounds which result in a full classification of the parameterized complexity of the problem with respect to natural structural parameterizations for the problem -- notably maximum capacity, treewidth, maximum degree, and maximum flow length. In particular, we obtain a fixed-parameter algorithm for the problem when parameterized by all four of these parameters, establish XP-tractability as well as W[1]-hardness with respect to the former three and latter three parameters, and all remaining cases remain paraNP-hard.