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In this paper, we consider the problem of designing cut sparsifiers andsketches for directed graphs. To bypass known lower bounds, we allow thesparsifier/sketch to depend on the balance of the input graph, which smoothlyinterpolates between undirected and directed graphs. We give nearly matchingupper 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 functionof cut balance, defined the maximum ratio of the cut value in the twodirections of a directed graph (Ene et al., STOC 2016). We also show aninteresting application of digraph sparsification via cut balance by using itto give a very short proof of a celebrated maximum flow result of Karger andLevine (STOC 2002).

Sparsification of Directed Graphs via Cut Balance