Approximation Algorithm for Directed Multicuts

Abstract

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${\{O({\sqrt n}), opt\}}$ 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 $\tilde O (n^{2/3}/opt^{1/3})$-approximation algorithm for UDM, which improves the ${\sqrt n}$-approximation for opt = ${\it \Omega}({\it n}^{\rm 1/2 + {\it \epsilon}}$). Combined with the paper of Gupta [5], we get that UDM can be approximated within better than $O({\sqrt n})$, unless $opt = \tilde\theta({\sqrt n})$. We also give a simple and fast O(n ^2/3)-approximation algorithm for DM.