Minimum Label s-t Cut has Large Integrality Gaps

TL;DR

This paper demonstrates that linear programming relaxations for the NP-hard Min Label s-t Cut problem have large integrality gaps, indicating limitations in designing approximation algorithms based solely on LP techniques.

Contribution

It proves significant integrality gaps for two LP formulations of the Min Label s-t Cut problem, showing inherent limitations of LP-based approximation methods.

Findings

Both LPs have large integrality gaps: Omega(m) and Omega(m^{1/3-epsilon}).

Results imply LP techniques alone cannot yield good approximation algorithms.

Highlights the complexity and difficulty of approximating Min Label s-t Cut.

Abstract

Given a graph G=(V,E) with a label set L = {l_1, l_2, ..., l_q}, in which each edge has a label from L, a source s in V, and a sink t in V, the Min Label s-t Cut problem asks to pick a set L' subseteq L of labels with minimized cardinality, such that the removal of all edges with labels in L' from G disconnects s and t. This problem comes from many applications in real world, for example, information security and computer networks. In this paper, we study two linear programs for Min Label s-t Cut, proving that both of them have large integrality gaps, namely, Omega(m) and Omega(m^{1/3-epsilon}) for the respective linear programs, where m is the number of edges in the graph and epsilon > 0 is any arbitrarily small constant. As Min Label s-t Cut 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 better approximation algorithms for Min Label s-t Cut that purely rely on linear programming.