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.

平衡旅行推销员问题（BTSP）是旅行推销员问题的一种变体，其中人们寻求一种旅行，使旅行中最大和最小边缘成本之间的差异最小化。BTSP 显然是 NP 难的，首先由 Larusic 和 Punnen 研究（Comput Oper Res 38(5):868–875, 2011）。他们提出了几种基于双阈值框架的启发式方法，这些方法收敛于高质量的解决方案，尽管并不总是最优的。在本文中，我们设计了一种专用的分支剪切算法来精确求解 BTSP。与经典的TSP相比，由于BTSP的目标函数，求解BTSP的算法的效率在很大程度上取决于正确确定路径中最大和最小的边缘成本。在所提出的分支切割算法中，我们开发了几种基于局部切割平面、边缘消除和变量固定的机制，以更精确地定位这些边缘成本。我们的方法中的其他关键成分是用于初始化 BTSP 最优值的下限和上限的算法，它们充当分支剪切算法的热启动。在 TSPLIB 实例的同一测试平台上进行的实验表明，我们的算法可以解决 65 个实例中的 63 个实例，并证明其最优性。