Exact Algorithms for the Quadratic Linear Ordering Problem

  • Authors:

  • Christoph Buchheim;Angelika Wiegele;Lanbo Zheng

  • Affiliations:

  • Forschungsinstitut für Diskrete Mathematik, Universität Bonn, 53113 Bonn, Germany;Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria;NICTA, The University of New South Wales, Sydney, New South Wales 2052, Australia

  • Venue:
  • INFORMS Journal on Computing
  • Year:
  • 2010

Quantified Score

Hi-index 0.01

Visualization

Abstract

The quadratic linear ordering problem naturally generalizes various optimization problems such as bipartite crossing minimization or the betweenness problem, which includes linear arrangement. These problems have important applications, e.g., in automatic graph drawing and computational biology. We present a new polyhedral approach to the quadratic linear ordering problem that is based on a linearization of the quadratic objective function. Our main result is a reformulation of the 3-dicycle inequalities using quadratic terms. After linearization, the resulting constraints are shown to be face-inducing for the polytope corresponding to the unconstrained quadratic problem. We use this result both within a branch-and-cut algorithm and within a branch-and-bound algorithm based on semidefinite programming. Experimental results for bipartite crossing minimization show that this approach clearly outperforms other methods.