Unsplittable Cost Flows from Unweighted Error-Bounded Variants

TL;DR

This paper investigates the relationship between two conjectures on converting fractional flows into unsplittable flows with bounded cost increases, providing new implications and a simplified proof technique that extends to related problems.

Contribution

It shows that Morell and Skutella's conjecture implies Goemans' conjecture with a bounded violation, and introduces a simple elementary proof technique applicable to various flow and network problems.

Findings

Morell and Skutella's conjecture implies Goemans' conjecture with a factor of two violation.

A simple elementary proof generalizes a technique for chain-constrained spanning trees.

The technique can transform cost-unaware algorithms into ones with cost guarantees in the weighted ring loading problem.

Abstract

A famous conjecture of Goemans on single-source unsplittable flows states that one can turn any fractional flow into an unsplittable one of no higher cost, while increasing the load on any arc by at most the maximum demand. Despite extensive work on the topic, only limited progress has been made. Recently, Morell and Skutella suggested an alternative conjecture, stating that one can turn any fractional flow into an unsplittable one without changing the load on any arc by more than the maximum demand. We show that their conjecture implies Goemans' conjecture (with a violation of twice the maximum demand). To this end, we generalize a technique of Linhares and Swamy, used to obtain a low-cost chain-constrained spanning tree from an algorithm without cost guarantees. Whereas Linhares and Swamy's proof relies on Langrangian duality, we provide a very simple elementary proof of a generalized version, which we hope to be of independent interest. Moreover, we show how this technique can also be used in the context of the weighted ring loading problem, showing that cost-unaware approximation algorithms can be transformed into approximation algorithms with additional cost guarantees.