New network decomposition theorems with applications

Network-decomposition theorems state that any undirected graph can be decomposed into small-diameter components by removing a small number of edges. Such theorems have proved useful in a variety of contexts, including synchronization of distributed systems, graph partitioning, and multicommodity flow. In this paper we present a decomposition theorem for directed graphs. We show how to combine techniques from multicommodity flow with this new decomposition to design an approximation algorithm for finding a minimum-cost directed multicut that separates a given set of node pairs. Computation of minimum cuts and multicuts is a basic step for approximation algorithms for a number of NP-complete problems. As an example, we show how to use our directed multicut approximation algorithm to approximately solve the minimum clause-deletion problem for 2-CNF formulae. We show that the new directed decomposition implies a polylogarithmic bound on the ratio between the capacity of the minimum multicut and the value of the maximum directed multicommodity flow in the case when the demands are symmetric. We also consider a generalization of the undirected decomposition theorem where instead of pairs of nodes, we consider sets of nodes. Roughly speaking, we prove that given a collection of node sets such that minimum-weight Steiner treemore » spanning each one of these sets is above a given threshold, one can remove a {open_quotes}small{close_quotes} number of edges such that in the resulting graph none of the specified sets will be wholly contained in a single connected component. This decomposition result is a corollary of a new polynomial-time polylogarithmic approximation algorithm for finding a minimum-weight multicut that separates all of the given node sets using multicommodity flow techniques.« less