Minimum Label s-t Cut has large integrality gaps

Abstract

The Min Labels-tCut problem is a fundamental problem in combinatorial optimization. This problem comes from many applications in real world, for example, information security and computer networks. We study two linear programs for Min Labels-tCut, proving that both of them have large integrality gaps, namely, Ω(m) and Ω(m1/3−ϵ) for the respective linear programs, where m is the number of edges in the input graph of the problem and ϵ>0 is any arbitrarily small constant. As Min Labels-tCut is NP-hard and the linear programming technique is a main approach to design approximation algorithms, our results give negative answer to the hope that designs can be found for better approximation algorithms for Min Labels-tCut that purely rely on linear programming.