Vertex Sparsification in Trees

Abstract

Given an unweighted tree \(T=(V,E)\) with terminals \(K \subset V\), we show how to obtain a 2-quality vertex flow and cut sparsifier H with \(V_H = K\). 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 \(V_H = K\) for such graphs. We then consider the other extreme and construct exact sparsifiers of size \(O(2^{k})\), when the input graph is unweighted.