 
Summary: Hardness of the Undirected Congestion Minimization Problem
Matthew Andrews
andrews@research.belllabs.com
Lisa Zhang
ylz@research.belllabs.com
Bell Laboratories
600700 Mountain Avenue
Murray Hill, NJ 07974
December 3, 2004
Abstract
We show that there is no (log log M) 1\Gamma'' approximation for the undirected congestion minimization
problem unless NP ` ZPT IME(n polylog n
), where M is the size of the graph and '' is any positive
constant.
1 Introduction
Consider a graph G with M edges and a set f(s i ; t i )g of sourcesink pairs. The congestion minimization
problem aims to connect all these pairs while minimizing the edge congestion, i.e. the maximum number
of demands that go through the same edge in G. The problem is known to be NPhard. The famous result
of Raghavan and Thompson [5] states that by applying randomized rounding to a linear relaxation of the
problem we obtain an O(log M= log log M) approximation for both directed and undirected graphs. On the
