Single-Source Unsplittable Flows in Planar Graphs

TL;DR

This paper proves that for planar graphs, a near-optimal unsplittable flow exists with at most double the capacity violation, advancing understanding of flow problems with capacity constraints.

Contribution

It establishes a new approximation result for planar graphs in the SSUF problem, connecting flow violations to discrepancy problems and confirming a conjecture for this class.

Findings

A 2-approximate unsplittable flow exists in planar graphs.

The approach reduces the number of paths per terminal iteratively.

Extends techniques to flows with bounds, confirming a conjecture.

Abstract

The single-source unsplittable flow (SSUF) problem asks to send flow from a common source to different terminals with unrelated demands, each terminal being served through a single path. One of the most heavily studied SSUF objectives is to minimize the violation of some given arc capacities. A seminal result of Dinitz, Garg, and Goemans showed that, whenever a fractional flow exists respecting the capacities, then there is an unsplittable one violating the capacities by at most the maximum demand. Goemans conjectured a very natural cost version of the same result, where the unsplittable flow is required to be no more expensive than the fractional one. This intriguing conjecture remains open. More so, there are arguably no non-trivial graph classes for which it is known to hold. We show that a slight weakening of it (with at most twice as large violations) holds for planar graphs. Our result is based on a connection to a highly structured discrepancy problem, whose repeated resolution allows us to successively reduce the number of paths used for each terminal, until we obtain an unsplittable flow. Moreover, our techniques also extend to simultaneous upper and lower bounds on the flow values. This also affirmatively answers a conjecture of Morell and Skutella for planar SSUF.