Sparsification of Directed Graphs via Cut Balance

TL;DR

This paper introduces a method for creating cut sparsifiers and sketches for directed graphs that adapt to the graph's balance, providing nearly optimal bounds and applications to maximum flow problems.

Contribution

It develops bounds for directed graph sparsification based on cut balance, bridging undirected and directed cases, and applies this to simplify maximum flow proofs.

Findings

Nearly matching upper and lower bounds for cut sparsifiers

Bounds depend on the graph's cut balance ratio

Application to simplified maximum flow proof

Abstract

In this paper, we consider the problem of designing cut sparsifiers and sketches for directed graphs. To bypass known lower bounds, we allow the sparsifier/sketch to depend on the balance of the input graph, which smoothly interpolates between undirected and directed graphs. We give nearly matching upper and lower bounds for both for-all (cf. Bencz\'ur and Karger, STOC 1996) and for-each (Andoni et al., ITCS 2016) cut sparsifiers/sketches as a function of cut balance, defined the maximum ratio of the cut value in the two directions of a directed graph (Ene et al., STOC 2016). We also show an interesting application of digraph sparsification via cut balance by using it to give a very short proof of a celebrated maximum flow result of Karger and Levine (STOC 2002).