The 2-opt, 3-opt or k–opt heuristics are classical local search algorithms for traveling salesman problems (TSP) in combinatorial optimization area, while sequential k–opt complete neighborhood examination takes polynomial time complexity which is timeconsuming to approach large scale TSP instances. This paper introduces a reasonable methodology called “multiple k–opt evaluation, multiple k–opt moves” that allows to simultaneously execute, without interference, massive k −opt moves that are globally found on the same TSP tour, as well as keep high performance GPU (Graphics Processing Unit) parallel 2-/3-opt evaluation with characteristic of “data parallelism, decentralized control and O(1) local memory for each GPU thread”. The methodology is reasonable since intervention of a sequential O(N) time complexity tour reversal operation is unavoidable for each k −opt move when using array of ordered coordinates as TSP tour data structure for high performance GPU k −opt local search that considers coalesced memory access and usage of limited on-chip shared memory. Innovation work includes two parts, a sequential non-interacted k-opt moves’ set partition algorithm that takes linear time complexity; a new TSP tour representation, array of ordered coordinates-index, that unveils how to combine the advantages of using doubly linked list and array of ordered coordinates data structures for iterative parallel k −opt local search based on GPU CUDA. We test this methodology on 22 national TSP instances with up to 71009 cities and with brute initial tour solution. Average maximum 997 non-interacted 2-opt moves are found and executed on the same tour of ch71009.tsp instance after one iteration of complete N∗(N−1)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac {N*(N-1)}{2}$\end{document} 2-opt checks working in parallel on GPU. And the proposed iterative GPU parallel 2-opt methodology executes average 306631 2-opt moves while only iterates 786 tour reversal operations, in comparison with methods that have to execute tour reversal operation after each 2-opt move. Experimental comparisons show that our proposed methodology gets huge acceleration over both classical sequential and a current state-of-the-art GPU parallel 2-opt implementation.

中文翻译：

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Multiple k -opt 评估多个 k -opt 使用 GPU 高性能本地搜索移动到大规模旅行商问题

2-opt、3-opt 或 k-opt 启发式算法是组合优化领域中旅行商问题 (TSP) 的经典局部搜索算法，而顺序 k-opt 完整邻域检查采用多项式时间复杂度，这对于接近大规模 TSP 是耗时的实例。本文介绍了一种称为“多个 k-opt 评估，多个 k-opt 移动”的合理方法，该方法允许在没有干扰的情况下同时执行在同一 TSP 巡视中全局发现的大量 k-opt 移动，并保持高性能GPU（图形处理单元）并行 2-/3-opt 评估，具有“数据并行性、分散控制和每个 GPU 线程的 O(1) 本地内存”的特点。该方法是合理的，因为对于考虑合并内存的高性能 GPU k -opt 局部搜索，当使用有序坐标数组作为 TSP 巡回数据结构时，对于每个 k -opt 移动都不可避免地需要干预顺序 O(N) 时间复杂度巡回反转操作访问和使用有限的片上共享内存。创新工作包括两部分，一个采用线性时间复杂度的顺序非交互k-opt移动的集合划分算法；一种新的 TSP 旅游表示，有序坐标索引数组，它揭示了如何结合使用双向链表和有序坐标数据结构数组的优势，用于基于 GPU CUDA 的迭代并行 k-opt 局部搜索。我们在多达 71009 个城市的 22 个国家 TSP 实例和粗暴的初始旅游解决方案上测试了这种方法。在一次完整的 N∗(N−1)2\documentclass[12pt]{minimal} \usepackage{amsmath} \ 迭代后，在 ch71009.tsp 实例的同一次游览中发现并执行平均最大 997 个非交互 2-opt 移动usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac {N* (N-1)}{2}$\end{document} 2-opt 检查在 GPU 上并行工作。与必须在每次 2-opt 移动后执行巡回逆转操作的方法相比，所提出的迭代 GPU 并行 2-opt 方法平均执行 306631 次 2-opt 移动，而仅迭代 786 次巡回逆转操作。