Sparsification of Balanced Directed Graphs

Sparsification, where the cut values of an input graph are approximately preserved by a sparse graph (called a cut sparsifier) or a succinct data structure (called a cut sketch), has been an influential tool in graph algorithms. But, this tool is restricted to undirected graphs, because some directed graphs are known to not admit sparsification. Such examples, however, are structurally very dissimilar to undirected graphs in that they exhibit highly unbalanced cuts. This motivates us to ask: can we sparsify a balanced digraph? To make this question concrete, we define balance $\beta$ of a digraph as the maximum ratio of the cut value in the two directions (Ene et al., STOC 2016). We show the following results: For-All Sparsification: If all cut values need to be simultaneously preserved (cf. Bencz\'ur and Karger, STOC 1996), then we show that the size of the sparsifier (or even cut sketch) must scale linearly with $\beta$. The upper bound is a simple extension of sparsification of undirected graphs (formally stated recently in Ikeda and Tanigawa (WAOA 2018)), so our main contribution here is to show a matching lower bound. For-Each Sparsification: If each cut value needs to be individually preserved (Andoni et al., ITCS 2016), then the situation is more interesting. Here, we give a cut sketch whose size scales with $\sqrt{\beta}$, thereby beating the linear lower bound above. We also show that this result is tight by exhibiting a matching lower bound of $\sqrt{\beta}$ on "for-each" cut sketches. Our upper bounds work for general weighted graphs, while the lower bounds even hold for unweighted graphs with no parallel edges.