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

Indonesian Journal of Electrical Engineering and Computer Science Vol. 25, No. 3, March 2022, pp. 1795~1802 ISSN: 2502-4752, DOI: 10.11591/ijeecs.v25.i3.pp1795-1802  1795 Parallelized solution to the asymmetric travelling salesman problem using central processing unit acceleration Akschat Arya1, Boominathan Perumal2, Santhi Krishnan3 1 BTech in Computer Science and Engineering, VIT University, Vellore, India 2 Department of Information Security, VIT University, Vellore, India 3 Department of Analytics, VIT University, Vellore, India Article Info ABSTRACT Article history: Travelling salesman problem is a well researched problem in computer science and has many practical applications. It is classified as a NP-hard problem as its exact solution can only be obtained in exponential time unless P = NP. There are different variants of the travelling salesman problem (TSP) and in this paper, asymmetric travelling salesman problem is addressed since this variant is quite often observed in real world scenarios. There are a number of heuristic approaches to this problem which provides approximate solutions in polynomial time, however this paper proposes an exact optimal solution which is accelerated with the help of multi-threadingbased parallelization. In order to find the exact optimal solution, we have used the held-karp algorithm involving dynamic programming and to reduce the time taken to find the optimal path, we have used a multi-threaded approach to parallelize the processing of sub-problems by leveraging the central processing unit cores (CPUs). This method is an extension of a well researched solution to the TSP; however, this method shows that solutions to computationally intensive problems involving sub-problems such as the asymmetic travelling salesman problem (ATSP) can be accelerated with the help of modern CPUs. Received Aug 18, 2021 Revised Jan 11, 2022 Accepted Jan 24, 2022 Keywords: Asymmetric travelling salesman problem Dynamic programming Held-karp algorithm Multithreading Parallelization This is an open access article under the CC BY-SA license. Corresponding Author: Akschat Arya BTech in Computer Science and Engineering, VIT University Vellore, India Email: 1. INTRODUCTION In the simple traveling salesperson problem (TSP), we are given an undirected graph 𝐺 = (𝑉, 𝐸) and 𝑐𝑜𝑠𝑡 𝑐(𝑒) > 0 for each edge 𝑒 ∈ 𝐸 and the objective is to find a hamiltonian cycle with the minimum cost. A hamiltonian cycle visits every vertex in 𝑉 exactly once. In this paper we are addressing the asymmetric travelling salesman problem (ATSP) which frequently has to be dealt with in real world scenarios. Let 𝑀 = (𝑉, 𝐴) be a given directed graph, with vertex set 𝑉 = {1, . . . , 𝑛} and arc set 𝐴 = {(𝑖, 𝑗) ∶ 𝑖, 𝑗 ∈ 𝑉}. Let 𝑐𝑖𝑗 be the cost for the arc (𝑖, 𝑗) ∈ 𝑉 with 𝑐𝑖𝑖 = +∞ (𝑖 ∈ 𝑉). A hamiltonian circuit (tour) of 𝐺 is a circuit visiting each vertex of 𝑉exactly once. The objective of the ATSP is to find a Hamiltonian circuit 𝑀 ∗ = (𝑉, 𝐴 ∗) of 𝑀 with minimum cost = ∑(𝑖,𝑗)∈𝐴∗ 𝑐𝑖𝑗 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 Journal homepage: http://ijeecs.iaescore.com 1796  ISSN: 2502-4752 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. CPU parallelization has also been achieved for other algorithms like the ant colony optimization for the TSP. Ling et al. in [5] have presented an adaptive parallel ant colony optimization (PACO) algorithm using massively parallel processors (MPPs). A method of adjusting the time interval adaptively for information exchange according to the diversity of the solutions is also proposed by Chen ling et al. to avoid early convergence and improve the quality of results [5]. Fejzagić et al. have shown that it is possible to efficiently parallelize metaheuristic algorithms like ACO using task parallel library [6]. Gizems Ermis et al. have investigated the acceleration from CUDA by using 2-opt and 3-opt local search heuristics and shared explained some parallelization strategies to utilize GPU resources effectively [7]. Haim Kaplan et al. has provided approximation algorithms for asymmetric TSP by the decomposition of directed regular multigraphs [2]. Experiments by Saxena et al. in [8] show that parallelization tools like OpenMP and CUDA can significantly reduce the execution time for genetic algorithms used in solving the TSP. Rashid in [9] presented a parallel heuristic integrating a greedy approach into a genetic algorithm with local-search using GPU acceleration. Most of the previous work have presented an approximate algorithm for the general TSP or an exact algorithm without CPU parallelization for the ASTP. In this paper we present an exact algorithm for the asymmetric TSP utilizing CPU parallelization and thread-safe hashmap to accelerate the execution process. Alrashdan et al. have used enhanced crossover operation using genetic algorithm with their probabilities in order to create an efficient method to provide a near optimal solution for the ATSP [10]. A Two-way parallel slime mold algorithm by flow and distance (TPSMA) is proposed by Liu et al. in [11] in order to solve slime mold algorithm’s problem of poor local optimization. Ascheuer et al. has provided a computational study which has indicated that most ATSP with time windows instances ranging till 50–70 nodes can be optimally solved using branch and cut [12]. Kang et al. propose an effective method of constructive crossover such that large number of genes can be effectively evolved by exploiting the GPUs parallel computing power and an effective parallel approach to genetic TSP where crossover methods cannot be easily implemented in parallel fashion [13]. Vasilchikov has shown that the little algorithm also has good (...truncated)