Cuts and Orderings: On Semidefinite Relaxations for the Linear Ordering Problem

Abstract

AbstractThe linear ordering problem is easy to state: Given a complete weighted directed graph, find an ordering of the vertices that maximizes the weight of the forward edges. Although the problem is NP-hard, it is easy to estimate the optimum to within a factor of 1/2. It is not known whether the maximum can be estimated to a better factor using a polynomial-time algorithm. Recently it was shown [NV01] that widely-studied polyhedral relaxations for this problem cannot be used to approximate the problem to within a factor better than 1/2. This was shown by demonstrating that the integrality gap of these relaxations is 2 on random graphs with uniform edge probability \(p = 2^{\sqrt{\log{n}}}/n\). In this paper, we present a new semidefinite programming relaxation for the linear ordering problem. We then show that if we choose a random graph with uniform edge probability \(p = \frac{d}{n}\), where d = ω(1), then with high probability the gap between our semidefinite relaxation and the integral optimal is at most 1.64.KeywordsDirected GraphRandom GraphSemidefinite RelaxationForward EdgeInteger Quadratic ProgramThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.