The Traveling Salesman Problem with Precedence Constraints and Binary Costs

Abstract

Translate Article

Given a set of n elements V = {1, 2,..., n} and a precedence relation < P ⊆ V × V defined on V, where i < P j or (i, j) ∈ < P means that i precedes j, let us consider the complete digraph G = (V, A) with no loops and with (j, i) ∉ A if and only if i < P j. Let be assigned a cost c ij ∈ {0, 1} to each arc (i, j) ∈ A with c ij = 0 if and only if (j, i) ∉ A. Let G′ be the digraph obtained from G adding a dummy vertex s, the Traveling Salesman Problem with Precedence Constraints and Binary Costs (TSP-PC-BC) is to find a hamiltonian tour of minimum cost in G′, starting from vertex s, visiting each vertex i ∈ V after having visited every vertex j ∈ V that must precede i and returning to vertex s. In this paper we investigate the TSP-PC-BC using a different approach from those used in previous works on TSP: we replace the graph G′ with a partially ordered set P = (V, < P ), defined by < P on V and reformulate the problem as the Jump Number Problem of P. We also propose exact and heuristic solution algorithms. Computational results on a set of randomly generated test problem are discussed.