The Black and White Traveling Salesman Problem

• Published in: Operations research Vol. 54; no. 2; pp. 366 - 378
• Main Authors: Ghiani, Gianpaolo, Laporte, Gilbert, Semet, Frederic
• Format: Journal Article
• Language: English
• Published: Linthicum, MD INFORMS 01.03.2006; Institute for Operations Research and the Management Sciences
• Subjects: Algorithms; Analysis; Applied sciences; branch-and-cut; Cardinality; Choice of transportation; exact algorithm; Exact sciences and technology; Graphs; Heuristics; Integer programming; Integers; Logistics; Operational research and scientific management; Operational research. Management science; Radiocarbon; Route optimization; Scheduling algorithms; Studies; Tabu search; Traveling salesman problem; valid inequalities; Vehicle capacity; vehicle routing problem; Vehicles; Vertices; Canada; Partition; traveling salesman problem; branch-and-cut; Vehicle routing problem; valid inequalities; Travelling salesman problem; Vertex(graph); Telecommunication; Scheduling; exact algorithm; Branch and cut method; Air transportation; Overhead line; Hamiltonian; Tour
• ISSN: 0030-364X; 1526-5463
• DOI: 10.1287/opre.1050.0218

The black and white traveling salesman problem (BWTSP) is defined on a graph G whose vertex set is partitioned into black and white vertices. The aim is to design a shortest Hamiltonian tour on G subject to cardinality and length constraints: both the number of white vertices as well as the length of the tour between two consecutive black vertices are bounded above. The BWTSP has applications in airline scheduling and in telecommunications. This paper proposes an integer linear formulation for the undirected BWTSP, as well as several classes of valid inequalities. An exact branch-and-cut algorithm is then developed. Extensive tests show that it can solve exactly instances involving up to 100 vertices. The algorithm can also be applied directly to solve unit demand vehicle routing problems of similar sizes.