Multiple k -opt evaluation multiple k -opt moves with GPU high performance local search to large-scale traveling salesman problems

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 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.