A branch-and-cut algorithm for the balanced traveling salesman problem

The balanced traveling salesman problem (BTSP) is a variant of the traveling salesman problem, in which one seeks a tour that minimizes the difference between the largest and smallest edge costs in the tour. The BTSP, which is obviously NP-hard, was first investigated by Larusic and Punnen (Comput Oper Res 38(5):868–875, 2011). They proposed several heuristics based on the double-threshold framework, which converge to good-quality solutions though not always optimal. In this paper, we design a special-purpose branch-and-cut algorithm for exactly solving the BTSP. In contrast with the classical TSP, due to the BTSP’s objective function, the efficiency of algorithms for solving the BTSP depends heavily on determining correctly the largest and smallest edge costs in the tour. In the proposed branch-and-cut algorithm, we develop several mechanisms based on local cutting planes, edge elimination, and variable fixing to locate those edge costs more precisely. Other critical ingredients in our method are algorithms for initializing lower and upper bounds on the optimal value of the BTSP, which serve as warm starts for the branch-and-cut algorithm. Experiments on the same testbed of TSPLIB instances show that our algorithm can solve 63 out of 65 instances to proven optimality.