Semidefinite relaxations of ordering problems

Abstract

Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element u is before v in the ordering. In the second case, the profit depends on whether u is before v and r is before s.The linear ordering problem is well studied, with exact solution methods based on polyhedral relaxations. The quadratic ordering problem does not seem to have attracted similar attention. We present a systematic investigation of semidefinite optimization based relaxations for the quadratic ordering problem, extending and improving existing approaches. We show the efficiency of our relaxations by providing computational experience on a variety of problem classes.

An SDP approach to multi-level crossing minimization

“…In summary, we get the following tractable relaxation of P QC , part of which (without the matrix cuts (3.13)) has been investigated in [2] for bipartite crossing minimization problems, in [1] for single-row layout problems, and, including (3.13), in [20] for general quadratic linear ordering problems.…”

A computational study and survey of methods for the single-row facility layout problem

“…Finally we relate the two SDP heuristics from [5] and [26] concerning their computational costs and practical performance.…”

“…Anjos and Yen [9] suggested an alternative SDP relaxation and achieved optimality gaps no greater than 5 % for large instances with up to 100 facilities. Recently Hungerländer and Rendl [26] proposed a general approach for quadratic ordering problems, where they further improved on the tightness of the above SDP relaxations. They used a suitable combination of optimization methods to deal with the stronger but more expensive relaxations and applied their method among others to some selected medium (SRFLP) instances.…”

“…Secondly we apply the approach from [26] for the first time to a broad selection of small, medium and large instances and compare it computationally to the leading algorithms for the different instance sizes. Thereby we demonstrate that this approach clearly dominates all other methods, permitting significant progress for medium as well as large instances.…”

“…In Section 2, we put the most competitive algorithms for (SRFLP) into perspective and compare them from a theoretical point of view. In Section 3, we conduct an extensive computational study for the SDP approach of Hungerländer and Rendl [26], achieving significant progress for medium and large instances. Finally some conclusions and current research are summarized in Section 4.…”