Edge disjoint paths revisited

The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the Ω(<i>m</i>^1/2-ε)-hardness result of Guruswami et al. [10] and the <i>O</i>(√<i>m</i>) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here <i>m</i> is the number of edges in the graph. However, we observe that the hardness of approximation shown in [10] applies to <i>sparse</i> graphs and hence when expressed as a function of <i>n</i>, the number of vertices, only an Ω(<i>n</i>^1/2-ε)-hardness follows. On the other hand, the <i>O</i>(√<i>m</i>)-approximation algorithms do not guarantee a <i>sub-linear</i> (in terms of <i>n</i>) approximation algorithm for <i>dense</i> graphs. We note that a similar gap exists in the known results on the integrality gap of the natural LP relaxation: an Ω(√<i>n</i>) lower bound and an <i>O</i>(√<i>m</i>) upper bound. Motivated by this discrepancy in the upper and lower bounds we study algorithms for the EDP in directed and undirected graphs obtaining improved approximation ratios. We show that the greedy algorithm has an approximation ratio of <i>O</i>(min(<i>n</i>^2/3, √<i>m</i>)) in undirected graphs and a ratio of <i>O</i>(min(<i>n</i>^4/5, √<i>m</i>)) in directed graphs. For ayclic graphs we give an <i>O</i>(√<i>n</i> log <i>n</i>) approximation via LP rounding. These are the first sub-linear approximation ratios for EDP. Our results also extend to EDP with weights and to the unsplittable flow problem with uniform edge capacities.