 
Summary: Hardness of the EdgeDisjoint Paths Problem with Congestion
Matthew Andrews
andrews@research.belllabs.com
Lisa Zhang
ylz@research.belllabs.com
Bell Laboratories
600700 Mountain Avenue
Murray Hill, NJ 07974
April 26, 2005
Abstract
In the EdgeDisjoint Paths problem with Congestion (EDPwC), we are given a graph with M edges
and a set of terminal pairs. The objective is to route as many terminal pairs as possible subject to the
constraint that at most w demands can be routed through any edge in the graph. In this paper, we study
the hardness of EDPwC in undirected graphs. We show that for any constant '' ? 0 and any conges
tion w = o(log log M= log log log M) there is no log
1\Gamma''
w+1 Mapproximation algorithm for EDPwC, unless
NP ` ZPT IME(n polylog n ). For larger congestions w Ÿ fl log log M= log log log M for some constant
fl, we obtain superconstant inapproximability ratios. Our reduction makes use of the Raz verifier and builds
upon the hardness for a related congestion minimization problem [1].
