Improved Lower Bounds on Multiflow-Multicut Gaps

TL;DR

This paper improves the known lower bounds on the ratio between minimum multicut and maximum multiflow in planar graphs from 2 to 16/7, introducing new techniques for such proofs.

Contribution

It establishes a tighter lower bound of 16/7 for planar graphs' multiflow-multicut gap and develops novel proof techniques applicable to related problems.

Findings

Multiflow-multicut gap in planar graphs is at least 16/7.

New proof techniques for lower bounds are introduced.

The result improves the longstanding bound of 2.

Abstract

Given a set of source-sink pairs, the maximum multiflow problem asks for the maximum total amount of flow that can be feasibly routed between them. The minimum multicut, a dual problem to multiflow, seeks the minimum-cost set of edges whose removal disconnects all the source-sink pairs. It is easy to see that the value of the minimum multicut is at least that of the maximum multiflow, and their ratio is called the multiflow-multicut gap. The classical max-flow min-cut theorem states that when there is only one source-sink pair, the gap is exactly one. However, in general, it is well known that this gap can be arbitrarily large. In this paper, we study this gap for classes of planar graphs and establish improved lower bound results. In particular, we show that this gap is at least $\frac{16}{7}$ for the class of planar graphs, improving upon the decades-old lower bound of 2. More importantly, we develop new techniques for proving such a lower bound, which may be useful in other settings as well.