The Sketching Complexity of Graph Cuts

We study the problem of sketching an input graph, so that, given the sketch, one can estimate the value (capacity) of any cut in the graph up to 1+ε approximation. Our results include both upper and lower bound on the sketch size, expressed in terms of the vertex-set size n and the accuracy ε. We design a randomized scheme which, given ε ∈ (0, 1) and an n-vertex graph G = (V,E) with edge capacities, produces a sketch of size Õ(n/ε) bits, from which the capacity of any cut (S, V \ S) can be reported, with high probability, within approximation factor (1 + ε). The previous upper bound is Õ(n/ε) bits, which follows by storing a cut sparsifier graph as constructed by Benczúr and Karger [BK96] and followup work [SS11, BSS12, FHHP11, KP12]. In contrast, we show that if a sketch succeeds in estimating the capacity of all cuts (S, S̄) in the graph (simultaneously), it must be of size Ω(n/ε) bits.