Approximating directed multicuts

The seminal paper of Leighton and Rao (1988) and subsequent papers presented approximate min-max theorems relating multicommodity flow values and cut capacities in undirected networks, developed the divide-and-conquer method for designing approximation algorithms, and generated novel tools for utilizing linear programming relaxations. Yet, despite persistent research efforts, these achievements could not be extended to directed networks, excluding a few cases that are "symmetric" and therefore similar to undirected networks. This paper is an attempt to remedy the situation. We consider the problem of finding a minimum multicut in a directed multicommodity flow network, and give the first nontrivial upper bounds on the max flow-to-min multicut ratio. Our results are algorithmic, demonstrating nontrivial approximation guarantees.