Vertex Sparsification and Oblivious Reductions

Abstract

Given an undirected, capacitated graph $G = (V, E)$ and a set $K \subset V$ of terminals of size $k$, we construct an undirected, capacitated graph $G' = (K, E')$ for which the cut function approximates the value of every minimum cut separating any subset $U$ of terminals from the remaining terminals $K - U$. We refer to this graph $G'$ as a cut-sparsifier, and we prove that there are cut-sparsifiers that can approximate all these minimum cuts in $G$ to within an approximation factor that depends only polylogarithmically on $k$, the number of terminals. We prove such cut-sparsifiers exist through a zero-sum game, and we construct such sparsifiers through oblivious routing guarantees. These results allow us to derive a more general theory of Steiner cut and flow problems, and allow us to obtain approximation algorithms with guarantees independent of the size of the graph for a number of graph partitioning, graph layout, and multicommodity flow problems for which such guarantees were previously unknown.