Improved Upper Bounds for the Directed Flow-Cut Gap

TL;DR

This paper improves the upper bounds on the directed flow-cut gap in graphs, narrowing the gap between known bounds and introducing new reductions among related problems.

Contribution

It establishes new upper bounds for the directed flow-cut gap and expands the network of reductions, showing near-equivalence of different problem variants.

Findings

Flow-cut gap for n-node directed graphs is at most n^{1/3 + o(1)}.

Upper bound on flow-cut gap of W^{1/2}n^{o(1)} where W is the sum of minimum fractional cut weights.

Near-equivalence between edge and vertex directed flow-cut gaps.

Abstract

We prove that the flow-cut gap for $n$-node directed graphs is at most $n^{1/3 + o(1)}$. This is the first improvement since a previous upper bound of $\widetilde{O}(n^{11/23})$ by Agarwal, Alon, and Charikar (STOC '07), and it narrows the gap to the current lower bound of $\widetilde{\Omega}(n^{1/7})$ by Chuzhoy and Khanna (JACM '09). We also show an upper bound on the directed flow-cut gap of $W^{1/2}n^{o(1)}$, where $W$ is the sum of the minimum fractional cut weights. As an auxiliary contribution, we significantly expand the network of reductions among various versions of the directed flow-cut gap problem. In particular, we prove near-equivalence between the edge and vertex directed flow-cut gaps, and we show that when parametrizing by $W$, one can assume unit capacities and uniform fractional cut weights without loss of generality.