Approximation algorithm 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 every (s, t) ∈ K. In the uncapacitated case (UDM) the goal is to find a minimum size K-multicut. The best approximation ratio known for DM is min{O(√n),opt} by Anupam Gupta [5], where n = |V|, and opt is the optimal solution value. All known non-trivial approximation algorithms for the problem solve large linear programs. We give the first combinatorial approximation algorithms for the problem. Our main result is a Ō(n^2/3/opt^1/3)- approximation algorithm for UDM, which improves the √n-approximation for opt = Ω(n^1/2+ε). Combined with the paper of Gupta [5], 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.