The Parameterised Complexity of Integer Multicommodity Flow

TL;DR

This paper explores the parameterised complexity of the Integer Multicommodity Flow problem, revealing its computational hardness under various parameters and identifying conditions under which it becomes fixed-parameter tractable.

Contribution

It provides a comprehensive complexity analysis of the general Integer Multicommodity Flow problem, extending beyond special cases and identifying new fixed-parameter tractability results.

Findings

XNLP-complete with pathwidth as parameter for unary capacities

NP-complete with binary capacities even for small pathwidth

Fixed-parameter tractability with weighted tree partition width as parameter

Abstract

The Integer Multicommodity Flow problem has been studied extensively in the literature. However, from a parameterised perspective, mostly special cases, such as the Disjoint Paths problem, have been considered. Therefore, we investigate the parameterised complexity of the general Integer Multicommodity Flow problem. We show that the decision version of this problem on directed graphs for a constant number of commodities, when the capacities are given in unary, is XNLP-complete with pathwidth as parameter and XALP-complete with treewidth as parameter. When the capacities are given in binary, the problem is NP-complete even for graphs of pathwidth at most 13. We give related results for undirected graphs. These results imply that the problem is unlikely to be fixed-parameter tractable by these parameters. In contrast, we show that the problem does become fixed-parameter tractable when weighted tree partition width (a variant of tree partition width for edge weighted graphs) is used as parameter.