Hardness of Directed Routing with Congestion
Julia Chuzhoy, Sanjeev Khanna · 2006
Given a graph G and a collection of source-sink pairs in G, what is the least integer c such that each source can be connected by a path to its sink, with at most c paths going through an edge? This is known as the congestion minimization problem, and the quantity c is called the congestion. Congestion minimization is one of the most well-studied NP-hard optimization problems. It is well-known that the elegant randomized rounding technique of Raghavan and Thompson can be used to obtain a solution with congestion at most c ∗ log n + O( log log n) where c ∗ is the optimal congestion. In this paper we show that there exists a δ> 0 such that no polynomial-time algorithm can guarantee a solution with congestion c ∗ δ log n + ( ) unless NP is log log n contained in ZPTIME(nlog log n). We also study the directed edge-disjoint paths (EDP) problem with congestion. The input