In recent years, there has been much interest in phase transitions of combinatorial problems. Phase transitions have been successfully used to analyze combinatorial optimization problems, characterize their typical-case features and locate the hardest problem instances. In this paper, we study phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an NP-hard combinatorial optimization problem that has many real-world applications. Using random instances of up to 1,500 cities in which intercity distances are uniformly distributed, we empirically show that many properties of the problem, including the optimal tour cost and backbone size, experience sharp transitions as the precision of intercity distances increases across a critical value. Our experimental results on the costs of the ATSP tours and assignment problem agree with the theoretical result that the asymptotic cost of assignment problem is pi ^2 /6 the number of cities goes to infinity. In addition, we show that the average computational cost of the well-known branch-and-bound subtour elimination algorithm for the problem also exhibits a thrashing behavior, transitioning from easy to difficult as the distance precision increases. These results answer positively an open question regarding the existence of phase transitions in the ATSP, and provide guidance on how difficult ATSP problem instances should be generated.
Solving the clustered traveling salesman problem with d-relaxed priority rule
University of Technology and Engineering, Vietnam National University（越南国立大学工程技术大学）；Interuniversitary Research Centre on Entreprise Networks, Logistics and Transportation (CIRRELT)（企业网络、物流与运输校际研究中心（CIRRELT））；Polytechnique Montréal（蒙特利尔理工学院）
Combining Reinforcement Learning with Lin-Kernighan-Helsgaun Algorithm for the Traveling Salesman Problem
School of Computer Science, Huazhong University of Science and Technology（华中科技大学计算机科学与技术学院）；Institute of Artificial Intelligence, Huazhong University of Science and Technology（华中科技大学人工智能研究院）；MIS, University of Picardie Jules Verne（皮卡第儒勒-凡尔纳大学MIS实验室）
Fixed set search applied to the traveling salesman problem
Qatar Environment and Energy Research Institute (QEERI), Hamad bin Khalifa University（哈马德·本·哈利法大学卡塔尔环境与能源研究所）；State University of Novi Pazar（新帕扎尔州立大学）；University of Hamburg（汉堡大学）；Pontificia Universidad Católica de Valparaíso（瓦尔帕莱索天主教大学）
Steiner Traveling Salesman Problem with Quantum Annealing
Department of Mechanical, Energy and Management Engineering, University of Calabria（机械、能源与管理工程系，卡塔尔大学）；TECNALIA, Basque Research and Technology Alliance (BRTA)（TECNALIA，巴斯克研究与技术联盟（BRTA））
