Edge-Disjoint Paths Revisited

The approximability of the maximum edge-disjoint paths problem (EDP) in directed graphs was seemingly settled by an Omega(m(1/2-epsilon))-hardness result of Guruswami et al. [2003], and an O(root m) approximation achievable via a natural multiconunodity-flow-based LP relaxation as well as a greedy algorithm. Here m is the number of edges in the graph. We observe that the Omega(m(1/2-epsilon))-hardness of approximation applies to sparse graphs, and hence when expressed as a function of n, that is, the number of vertices, only an Omega(n(1/2-epsilon))-hardness follows. On the other hand, O(root m)-approximation algorithms do not guarantee a sublinear (in terms of n) approximation algorithm for dense graphs. We note that a similar gap exists in the known results on the integrality gap of the flow-based LP relaxation: an Omega(root n) lower bound and O(root m) upper bound. Motivated by this discrepancy in the upper and lower bounds, we study algorithms for EDP in directed and undirected graphs and obtain improved approximation ratios. We show that the greedy algorithm has an approximation ratio of O(min(n(2/3), root m)) in undirected graphs and a ratio of O(min(n(4/5), root m)) in directed graphs. For acyclic graphs we give an O(root n ln n) approximation via LP rounding. These are the first sublinear approximation ratios for EDP. The results also extend to EDP with weights and to the uniform-capacity unsplittable flow problem (UCUFP).