A Polynomial Time Approximation Algorithm for the Two-Commodity Splittable Flow Problem

TL;DR

This paper introduces a polynomial-time 1/2-approximation algorithm for a generalized two-commodity flow problem where commodities can be split into chunks, addressing NP-hardness and providing exact solutions under certain conditions.

Contribution

It presents the first polynomial-time approximation algorithm for the two-commodity splittable flow problem with uniform chunk sizes and extends results to approximate maximum concurrent flows.

Findings

NP-hardness of the generalized problem

A polynomial-time 1/2-approximation algorithm for uniform chunk sizes

Exact solutions possible under specific cut conditions

Abstract

We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity $i\in{1,2}$ can be split into a bounded number $k_i$ of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of $\alpha>1/2$. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even $k_i$ and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent $(k_1,k_2)$-splittable flow without chunk size restrictions for fixed demand ratios.