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

Authors:
Michael Saks;Alex Samorodnitsky;Leonid Zosin
Affiliations:
Rutgers University, Department of Mathematics, 110 Frelinghuysen Road, 08854-8019, Piscataway, NJ, USA;The Hebrew University of Jerusalem, School of Computer Science and Engineering, 110 Frelinghuysen Road, 93708, Jerusalem, NJ, Israel;ITG Inc., School of Computer Science and Engineering, 44 Farnsworth Str., 02210, Boston, MA, USA
Venue:
Combinatorica
Year:
2004

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Abstract

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.