A Cutting Plane Algorithm for the Linear Ordering Problem

Abstract

The linear ordering problem is an NP-hard combinatorial optimization problem with a large number of applications (including triangulation of input-output matrices, archaeological senation, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences). In a former paper, we have investigated the facet structure of the 0/1-polytope associated with the linear ordering problem. Here we report on a new algorithm that is based on these theoretical results. The main part of the algorithm is a cutting plane procedure using facet defining inequalities. This procedure is combined with various heuristics and branch and bound techniques. Our computational results compare favorably with the results of existing codes. In particular, we could triangulate all input-output matrices, of size up to 60 × 60, available to us within acceptable time bounds.

A benchmark library and a comparison of heuristic methods for the linear ordering problem

“…This is a well-known set of instances, first used in [18]. It contains 50 real-world linear ordering problems taken from input-output tables from various sources.…”

The linear ordering problem: an algorithm for the optimal solution

“…However, it has been shown that this problem has 2 n       equations, n(n-1) nonnegativity constraints, 2 3 n       3-dicycle inequalities, 120 6 n       3-fence inequalities, and 360 6 n       Möbius ladder inequalities [6]. Due to this enormous number of constraints, it is impractical to list all the constraints and solve the linear program using available computer code.…”

“…The following heuristic was described by Reinelt [6]. The heuristic is based on the uniqueness of the solution in Figure 1 for any 3-fence.…”

Designing a Hybrid Genetic Algorithm for the Linear Ordering Problem

“…The Linear Ordering Problem(LOP) has numerous applications in economics, archaeology, scheduling, the social sciences, and aggregation of individual preferences [5,6,7,8]. Of all these applications, the most famous one may be "the Triangulation for Input-Output Matrices" [7], which measures the movement of goods from one "sector" to another in economics research.…”

“…Grotschel et al first proposed a cutting plane algorithm in [6,7]; Mitchell and Borchers [8] improved the result by combining the cutting plane with the interior point algorithm. However, all of these exact algorithms are extremely time-consuming.…”