Before revision: 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 graphsare 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.
After revision: 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.