Problema do arranjo linear mínimo

Abstract

Let G = (V, E) be a simple and undirected graph of set of vertices V and set of edges E. Given an assignment of distinct labels in {1, · · · , |V |} to the vertices of G, for every edge uv ∈ E, we define its weight as the absolute difference of labels given to its end nodes. 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. MinLA is an NP-Hard problem whose corresponding polyhedron has a factorial number of extreme points. In this work, we investigate a recent quadratic model for the MinLA presenting O(|V |² ) variables and O(|V |² ) constraints. This model has the smallest number of variables and constraints among all models in the literature for the problem. We present some theoretical results for the quadratic model, and show how to obtain a mixed linear model whose optimal solution is the same of the quadratic one. We also present valid inequalities for the proposed models. We discuss two heuristics for the problem and propose a column generation algorithm and a specialized Branch and Bound for MinLA. 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.