Accelerating the Held-Karp Algorithm for the Symmetric Traveling Salesman Problem

Abstract

In this paper, we propose an acceleration method for the Held-Karp algorithm that solves the symmetric traveling salesman problem by dynamic programming. The proposed method achieves acceleration with two techniques. First, we locate data-independent subproblems so that the subproblems can be solved in parallel. Second, we reduce the number of subproblems by a meet in the middle (MITM) technique, which computes the optimal path from both clockwise and counterclockwise directions. We show theoretical analysis on the impact of MITM in terms of the time and space complexities. In experiments, we compared the proposed method with a previous method running on a single-core CPU. Experimental results show that the proposed method on an 8-core CPU was 9.5-10.5 times faster than the previous method on a single-core CPU. Moreover, the proposed method on a graphics processing unit (GPU) was 30-40 times faster than that on an 8-core CPU. As a side effect, the proposed method reduced the memory usage by 48%. key words: symmetric traveling salesman problem, Held-Karp algorithm, parallelization, meet in the middle, GPU

Parallelized solution to the asymmetric travelling salesman problem using central processing unit acceleration

“…There are different variants of the travelling salesman problem which have been addressed by researchers earlier and both approximate (faster) and exact (slower) solutions have been provided. Some possible solutions for some of the other variants as per earlier research are as follows: i) symmetric TSP: GPU accelerated solution provided by Kimura et al in [1], ii) ATSP: approximation algorithms by decomposing directed regular multigraphs provided by Kaplan et al in [2], iii) ATSP with windows: exact solution through a graph transformation provided by Albiach et al in [3], iv) ATSP with replenishment arcs: polyhedral results provided by Mak and Boland in [4].…”

“…Meet in the middle algorithm was used by Kazuro Kimura et al to accelerate the execution time but this method can only be used on the symmetric TSP by leveraging the symmetric aspect of the problem and thus Kimura et al in [1] achieved an acceleration by a factor of 1.5 and that of 1.7 using man-in-the-middle (MITM) when n (number of vertices) was odd and even, respectively. Since this paper aims to address the asymmetric travelling salesman problem, we have not used MITM, instead we make use of the following techniques to accelerate the processing time: i) multi-threaded program to utilize central processing unit (CPU) cores, ii) thread-safe hashmap to store results of the dynamic cost function.…”