A New Hybrid Heuristic for Large Geometric Traveling Salesman Problems Based on the Delaunay Triangulation

In this paper we present a new hybrid heuristic for large geometric Traveling Salesman Problems. The method is based on the use of the Delaunay Triangulation of a set of cities. Many heuristics for the TSP are generally composed of two phases: a constructive heuristic followed by an iterative improvement step until a local minimum is reached. We think of this heuristic as a hybrid since, after a city has been introduced, a period of local improvement takes us to another local minimum before a new city is added. We provide computational results using many TSPLIB geometric instances. RESUMO: Neste artigo, apresentamos uma nova heur stica h brida para os problemas geom etrico do Caixeiro Viajante de grande porte. O m etodo e baseado no uso de \Delaunay Triangulation" de um conjunto de cidades. Muitas heur sticas para o problema do Caixeiro Viajante s~ ao geralmente compostas por duas fases: uma heur stica de contruu c~ ao e um passo de melhoria, at e que um otimo local seja alcann cado. NN os pensamos nesta heur stica como h brida desde que, depois que uma cidade e introduzida, um per odo de melhoria local nos leva a outro m nimo local antes que uma nova cidade tenha sido adicionada. Fornecemos resultados computacionais usando inst^ ancias do TSPLIB. The Voronoi diagram solves the \loci of proximity" problem: \Given a set S of N points in the plane, for each point p in S ; which is the locus of points (x; y) in the plane that are closer to p than to any other point of S ?". Given two points p and q, the set of points closer to p than to q is just the half-plane containing p that is deened by the perpendicular bisector of pq. Let us denote this half-plane by H (p; q). The Voronoi polygon associated to a point p (denoted V R (p)) is a convex polygonal region having no more than N ?1 sides since it is the intersection of N ?1 bisecting half-planes deened by p and the other N ? 1 points in S. Then V R (p) = \ p;q2S;p6 =q H (p; q) (1) Given the Voronoi diagram of S there is a graph which is its straight-line dual. It is called the Delaunay Triangulation (DT) denoting it as (S; D T). It is a graph that has S …