Greedy approximation algorithms for Directed Multicuts

The Directed Multicut (DM) problem is: given a simple directed graph G = (V, E) with positive capacities u _e on the edges, and a set K ⊆ V × V of ordered pairs of nodes of G, find a minimum capacity K-multicut; C ⊆ E is a K-multicut if in G - C there is no (s, t)-path for any (s, f) ε K. In the uncapacitated case (UDM) the goal is to find a minimum size K-multicut. The best approximation ratio known for DM is O(min{√n, opt}) by Gupta, where n = |V|, and opt is the optimal solution value. All known nontrivial approximation algorithms for the problem solve large linear programs. We give the first combinatorial approximation algorithms for the problem. Our main result is an Õ(n ^2/3/opt ^1/3|-approximation algorithm for UDM, which improves the Õ(min{opt, √n})-approximation for opt = Ω(n ^1/2+ε). Combined with the article of Gupta, we get that UDM can be approximated within better than O(√n), unless opt = Θ̃(√n). We also give a simple and fast O(n ^2/3)- approximation algorithm for DM.