A Lower Bound On The Integrality Gap For Minimum Multicut In Directed Networks

Given a directed edge-weighted graph and k source-sink pairs, the Minimum Directed Multicut Problem is to find an edge subset with minimal weight, that separates each source-sink pair. Determining the minimum multicut in directed or undirected graphs is NP-hard. The fractional version of the minimum multicut problem is dual to the maximum multicommodity flow problem. The integrality gap for an instance of this problem is the ratio of the minimum weight multicut to the minimum weight fractional multicut; trivially this gap is always at least 1 and it is easy to show that it is at most k. In the analogous problem for undirected graphs this upper bound was improved to O(log k).

In this paper, for each k an explicit family of examples is presented each with k source-sink pairs for which the integrality gap can be made arbitrarily close to k. This shows that for directed graphs, the trivial upper bound of k can not be improved.

This is a preview of subscription content, access via your institution.

Author information



Corresponding author

Correspondence to Michael Saks*.

Additional information

* This work was supported in part by NSF grant CCR-9700239 and by DIMACS.

† This work was done while a postdoctoral fellow at DIMACS.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Saks*, M., Samorodnitsky†, A. & Zosin†, L. A Lower Bound On The Integrality Gap For Minimum Multicut In Directed Networks. Combinatorica 24, 525–530 (2004). https://doi.org/10.1007/s00493-004-0031-x

Download citation

Mathematics Subject Classification (2000):

  • 05C20
  • 90B10
  • 90C35