Minimum Linear Arrangements

Let G = (V, E) be a simple undirected graph. Given distinct labels in {1,⋯,|V|} to the vertices of G, we define the weight of an edge uv∈E as the absolute difference between the labels assigned to u and v. The minimum linear arrangement problem (MinLA) consists in finding a labeling of the vertices of G such that the sum of the weights of its edges is minimized. It is an NP-Hard problem whose corresponding polyhedron has a factorial number of extreme points. We propose a quadratic model for MinLA and use it to obtain a novel compact mixed integer linear programming (MILP) formulation for the problem, featuring O(|V|2) variables and O(|V|2) constraints. We show the correctness of the new model and discuss valid inequalities for the problem. Our findings open new insights on the study of effective exact approaches to the problem. Computational experiments show that the new quadratic and mixed linear models performed better than existing ones in the literature for new and benchmark instances of this problem.