Cheeger-Type Approximation for Sparsest st-Cut

Robert Krauthgamer, Tal Wagner

Abstract: We introduce the st-cut version of the sparsest-cut problem, where the goal is to find a cut of minimum sparsity in a graph G(V, E) among those separating two distinguished vertices s, t ∈ V. Clearly, this problem is at least as hard as the usual (non-st) version. Our main result is a polynomial-time algorithm for the product-demands setting that produces a cut of sparsity O(√OPT), where OPT ⩽ 1 denotes the optimum when the total edge capacity and the total demand are assumed (by normalization) to be 1. Our result generalizes the recent work of Trevisan [arXiv, 2013] for the non-st version of the same problem (sparsest cut with product demands), which in turn generalizes the bound achieved by the discrete Cheeger inequality, a cornerstone of Spectral Graph Theory that has numerous applications. Indeed, Cheeger’s inequality handles graph conductance, the special case of product demands that are proportional to the vertex (capacitated) degrees. Along the way, we obtain an O(log |V|) approximation for the general-demands setting of sparsest st-cut.