Vertex sparsification in trees

Abstract

Given an unweighted tree T = (V,E) with terminals K ⊂ V, we show how to obtain a 2-quality vertex flow and cut sparsifier H with VH = K. We prove that our result is essentially tight by providing a 2− o(1) lower-bound on the quality of any cut sparsifier for stars. In addition we give improved results for quasi-bipartite graphs. First, we show how to obtain a 2-quality flow sparsifier with VH = K for such graphs. We then consider the other extreme and construct exact sparsifiers of size O(2k), when the input graph is unweighted.