New Hardness Results for Undirected Edge Disjoint Paths

In the edge-disjoint paths (EDP) problem, we are given a graph G and a set of source-sink pairs in G. The goal is connect as many pairs as possible in an edge-disjoint manner. This problem is NP-hard and the best known approximation algorithm gives an Õ(min{n2/3,√m})-approximation for both directed and undirected graphs; here n and m denote the number of vertices and edges in G respectively. For directed graphs, this result is tight as a function of m since it is known that directed EDP is NP-hard to approximate to within Ω(m ) for any > 0. However, for undirected graphs, until recently nothing better than APX-hardness was known. In a significant improvement, Andrews and Zhang [1] showed that undirected EDP is Ω(log n)-hard to approximate unless NP is contained in ZPTIME(n). In this paper, we improve the hardness result of [1] as well as obtain the first polylogarithmic integrality gaps and hardness results for undirected EDP when congestion is allowed. A solution to EDP has congestion c if we allow up to c paths to share an edge. When no congestion is allowed, we establish an Ω(log n)-hardness for EDP. With congestion c, we show that the natural multicommodity flow relaxation of EDP has an Ω(( log n (log log n)2 )/c) integrality gap. Finally, we show that it is possible to obtain a hardness result that is comparable to the integrality gap. In particular, we show that EDP is Ω ( (log n) )/( 3 2 c+ 1 2 ) ) -hard to approximate for any constant > 0, when congestion c is allowed, for any c = o(log log n)/(log log log n), such that c = 2 − 1 for some integer z. We also obtain super-constant hardness when c is as large as O(log log n)/(log log log n). Similar results can be obtained for the All-or-Nothing flow problem, a relaxation of EDP in that the unit flow between each routed source-sink pair does not have to be on a single path. Using standard transformations, these results can also be extended to the node-disjoint versions of these problems as well as to the directed setting.