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

Abstract

The 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.