Multicommodity Flow in Polynomial Time

TL;DR

This paper demonstrates that the multicommodity flow problem, previously known to be NP-hard, can be solved in polynomial time under certain broad conditions using advanced integer programming techniques.

Contribution

It introduces a novel application of n-fold integer programming to establish polynomial time solvability for specific cases of the multicommodity flow problem.

Findings

Polynomial time solvability in certain cases

Application of n-fold integer programming to network flow

Surprising tractability results for a previously NP-hard problem

Abstract

The multicommodity flow problem is NP-hard already for two commodities over bipartite graphs. Nonetheless, using our recent theory of n-fold integer programming and extensions developed herein, we are able to establish the surprising polynomial time solvability of the problem in two broad situations.