Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs

In the undirected Edge-Disjoint Paths problem with Congestion (EDPwC), we are given an undirected graph with V nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c = 1, the problem is simply referred to as the Edge-Disjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs. Our main result is that for every ε > 0 there exists an α > 0 such that for 1 6 c 6 α log log V log log log V , it is hard to distinguish between instances where we can route all terminal pairs on edge-disjoint paths, and instances where we can route at most a 1/(logV ) 1−ε c+2 fraction of the terminal pairs, even if we allow congestion c. This implies a (logV ) 1−ε c+2 hardness of approximation for EDPwC and an Ω(log log V/ log log logV ) hardness of approximation for the undirected congestion minimization problem. These results hold assuming NP 6⊆ ∪dZPTIME(2 d ). In the case that we do not require perfect completeness, i.e. we do not require that all terminal pairs are routed for “yes-instances”, we can obtain a slightly better inapproximability ratio of (log V ) 1−ε c+1 . Note that by setting c = 1 this implies that the regular EDP problem is (logV ) 1 2 −ε hard to approximate. Using standard reductions, our results extend to the node-disjoint versions of these problems as well as to the directed setting. We also show a (log V ) 1−ε c+1 inapproximability ratio for the All-or-Nothing Flow with Congestion (ANFwC) problem, a relaxation of EDPwC, in which the flow unit routed between the source-sink pairs does not have to follow a single path, so the resulting flow is not necessarily integral. Bell Laboratories, Lucent Technologies, Murray Hill, NJ. {andrews,ylz}@research.bell-labs.com Toyota Technological Institute, Chicago, IL 60637. Email: [email protected] Department of Computer Science and Engineering, University of Washington, Seattle, WA 98195: [email protected]. Supported by NSF CCF-0343672 and a Packard fellowship. Dept. of CIS, University of Pennsylvania, Philadelphia PA. Email: [email protected]. Supported in part by an NSF Career Award CCR-0093117 and by NSF Award CCF-0635084. Microsoft Research, Silicon Valley Campus, Mountain View, CA 94043. Email: [email protected] Electronic Colloquium on Computational Complexity, Report No. 113 (2007)