Large Traveling Salesman Problem Research Articles

An Efficient Tour Construction Heuristic for Generating the Candidate Set of the Traveling Salesman Problem with Large Sizes

In this paper, we address the challenge of creating candidate sets for large-scale Traveling Salesman Problem (TSP) instances, where choosing a subset of edges is crucial for efficiency. Traditional methods for improving tours, such as local searches and heuristics, depend greatly on the quality of these candidate sets but often struggle in large-scale situations due to insufficient edge coverage or high time complexity. We present a new heuristic based on fuzzy clustering, designed to produce high-quality candidate sets with nearly linear time complexity. Thoroughly tested on benchmark instances, including VLSI and Euclidean types with up to 316,000 nodes, our method consistently outperforms traditional and current leading techniques for large TSPs. Our heuristic’s tours encompass nearly all edges of optimal or best-known solutions, and its candidate sets are significantly smaller than those produced with the POPMUSIC heuristic. This results in faster execution of subsequent improvement methods, such as Helsgaun’s Lin–Kernighan heuristic and evolutionary algorithms. This substantial enhancement in computation time and solution quality establishes our method as a promising approach for effectively solving large-scale TSP instances.

Focused ACO with node relocation procedure for solving large TSP instances

Ant Colony Optimization (ACO) is a well-known family of nature-inspired metaheuristics used to find approximate solutions for challenging optimization problems. Despite its relative simplicity and adaptability to new problem variants, the efficiency of ACO significantly deteriorates with increasing problem size, even though it often provides high-quality solutions. This paper aims to enhance the performance of ACO when solving large instances of the Traveling Salesman Problem (TSP). Specifically, we propose novel improvements to the current state-of-the-art ACO approach, Focused ACO (FACO), that include a node relocation procedure applied during the solution construction process and a greedy heuristic that preserves high-quality solutions from previous algorithm iterations. Our proposed FACO variants demonstrate a substantial performance improvement when solving TSP instances with up to 200k nodes.

A New Constructive Heuristic Driven by Machine Learning for the Traveling Salesman Problem

Recent systems applying Machine Learning (ML) to solve the Traveling Salesman Problem (TSP) exhibit issues when they try to scale up to real case scenarios with several hundred vertices. The use of Candidate Lists (CLs) has been brought up to cope with the issues. A CL is defined as a subset of all the edges linked to a given vertex such that it contains mainly edges that are believed to be found in the optimal tour. The initialization procedure that identifies a CL for each vertex in the TSP aids the solver by restricting the search space during solution creation. It results in a reduction of the computational burden as well, which is highly recommended when solving large TSPs. So far, ML was engaged to create CLs and values on the elements of these CLs by expressing ML preferences at solution insertion. Although promising, these systems do not restrict what the ML learns and does to create solutions, bringing with them some generalization issues. Therefore, motivated by exploratory and statistical studies of the CL behavior in multiple TSP solutions, in this work, we rethink the usage of ML by purposely employing this system just on a task that avoids well-known ML weaknesses, such as training in presence of frequent outliers and the detection of under-represented events. The task is to confirm inclusion in a solution just for edges that are most likely optimal. The CLs of the edge considered for inclusion are employed as an input of the neural network, and the ML is in charge of distinguishing when such edge is in the optimal solution from when it is not. The proposed approach enables a reasonable generalization and unveils an efficient balance between ML and optimization techniques. Our ML-Constructive heuristic is trained on small instances. Then, it is able to produce solutions—without losing quality—for large problems as well. We compare our method against classic constructive heuristics, showing that the new approach performs well for TSPLIB instances up to 1748 cities. Although ML-Constructive exhibits an expensive constant computation time due to training, we proved that the computational complexity in the worst-case scenario—for the solution construction after training—is O(n2logn2), n being the number of vertices in the TSP instance.

A Multi Ant Colony Optimization Approach For The Traveling Salesman Problem

In this paper, we propose a new approach to solving the Traveling Salesman Problem (TSP), for which no exact algorithm is known that allows to find a solution in polynomial time. The proposed approach is based on optimization by ants. It puts several colonies in competition for improved solutions (in execution time and solution quality) to large TSP instances, and allows to efficiently explore the range of possible solutions. The results of our experiments show that the approach leads to better results compared to other heuristics from the literature, especially in terms of the quality of solutions obtained and execution time.

An Adaptive Ant Colony System Memorizing Better Solutions (aACS-MBS) For Traveling Salesman Problem

Choosing the optimal among the many alternatives that meet the criteria is one of the problems that occupy life. This kind of problems frequently encountered by commercial companies in daily life is one of the issues that operators focus on with care. Many techniques have been developed that can provide acceptable solutions in a reasonable time. However, one of the biggest problems for these techniques is that the appropriate values can be assigned to the algorithm parameters. Because one of the most important issues determining algorithm performance is the values to be assigned to its parameters. The Ant Colony System (ACS) is a metaheuristic method that produces successful solutions, especially in combinatorial optimization problems. However, it is very difficult to be able to direct the algorithm to different areas of the search space and, on the other hand, to maintain its local search capability. In this study, a solution proposal is presented that updates the q0 parameter dynamically, which balances the exploitation and exploration activities of the ACS. The method has been tested on the traveling salesman problem (TSP) of different sizes, and the obtained results are evaluated together with the change in the q0 parameter, and the solution search strategy of the algorithm is analyzed. With the pheromone maps formed as a result of the search, the effect of transfer functions was evaluated. Results obtained with aACS-MBS were compared with different ant colony optimization (ACO) algorithms. The aACS-MBS fell behind the most successful solution found in the literature, by up to 4%, in large TSP benchmarks. As a result, it has been seen that the method can be successfully applied to combinatorial optimization problems.

Generalize a Small Pre-trained Model to Arbitrarily Large TSP Instances

For the traveling salesman problem (TSP), the existing supervised learning based algorithms suffer seriously from the lack of generalization ability. To overcome this drawback, this paper tries to train (in supervised manner) a small-scale model, which could be repetitively used to build heat maps for TSP instances of arbitrarily large size, based on a series of techniques such as graph sampling, graph converting and heat maps merging. Furthermore, the heat maps are fed into a reinforcement learning approach (Monte Carlo tree search), to guide the search of high-quality solutions. Experimental results based on a large number of instances (with up to 10,000 vertices) show that, this new approach clearly outperforms the existing machine learning based TSP algorithms, and significantly improves the generalization ability of the trained model.

ACOTSP-MF: A memory-friendly and highly scalable ACOTSP approach

Ant Colony Optimization (ACO) is a population-based meta-heuristic inspired by the social behavior of ants. It is successfully applied in solving many NP-hard problems, such as the Traveling Salesman Problem (TSP). Large-sized instances pose two memory problems to the ACOTSP algorithm: the memory size and the memory bandwidth.This work has focused on developing ACOTSP-MF, a new ACOTSP algorithm proposed to adequately manage the memory issues that arise while solving large TSP instances. ACOTSP-MF uses the nearest neighbor list, introducing a novel class of cities, the backup cities, while grouping cities into three classes: the nearest neighbor cities, the backup cities, and the rest of the cities (the majority). ACOTSP-MF also modifies how the base ACOTSP carries out the tour construction and pheromone update phases depending on the group to which a city belongs. This way, ACOTSP-MF reduces both the memory requirements of its data structures (from O(n∗n) to O(n)), and the memory bandwidth needs (thanks to better exploitation of the memory data locality).In this paper, we have carried out an in-depth analysis of ACOTSP-MF performance for medium and large TSP instances, covering vectorization and scalability issues and showing its main bottlenecks. For medium-size instances, the paper reports speedup factors of 20-500X for the rl11849 instance compared to the base ACOTSP version. ACOTSP-MF is intended and especially adequate for large-size instances. In this context, the paper reports excellent execution time for the Tour Construction phase, with less than 500 ms per iteration for the earring-200k instance. Finally, a study about the solution quality of ACOTSP-MF has been included, showing that ACOTSP-MF paired with local search offers high solution quality (within 2% of the best-known solution).

Pearson correlation coefficient-based pheromone refactoring mechanism for multi-colony ant colony optimization

To solve the problem of falling into local optimum and poor convergence speed in large Traveling Salesman Problem (TSP), this paper proposes a Pearson correlation coefficient-based Pheromone refactoring mechanism for multi-colony Ant Colony Optimization (PPACO). First, the dynamic guidance mechanism is introduced to dynamically adjust the pheromone concentration on the path of the maximum and minimum spanning tree, which can effectively balance the diversity and convergence of the algorithm. Secondly, the frequency of communication between colonies is adjusted adaptively according to a criterion based on the similarity between the minimum spanning tree and the optimal solution. Besides, the pheromone matrix of the colony is reconstructed according to the Pearson correlation coefficient or information entropy to help the algorithm jump out of the local optimum, thus improving the accuracy of the solution. These strategies greatly improve the adaptability of the algorithm and ensure the effectiveness of the interaction. Finally, the experimental results indicate that the proposed algorithm could improve the solution accuracy and accelerate the convergence speed, especially for large-scale TSP instances.

An enhanced Moth-flame optimization algorithm for permutation-based problems

Moth-flame optimizer (MFO) is one of the recently proposed metaheuristic optimization techniques which has been successfully used in wide range of applications. However, there are two issues with the MFO algorithm. First, as a stochastic technique, MFO may prematurely converge at some local minima during the search process. Second, the original MFO was developed for continuous search space problems and is not directly applicable to, e.g., permutation-based problems (PBP). In this paper, a novel perturbation strategy is introduced to the MFO algorithm to avoid probable local minima regions. This strategy works as follows: if the best solution obtained so far doesn’t improve for a given number of consecutive iterations, the current population of solutions is perturbed using some crossover mechanism as an attempt to explore new promising neighbourhoods in the search space. In addition, smallest position values mapping technique is employed in order for the proposed, termed CrossMFO (COMFO), algorithm to be applicable to PBP problems. It is noticed that, despite these modifications, the proposed COMFO has the same time complexity order as the original MFO. Extensive simulation experiments are conducted to compare the proposed COMFO to the MFO, other enhanced versions of MFO, and some metaheuristic optimizers in solving the well-known Travelling Salesman Problem (TSP). Empirical results show that the solutions obtained using MFO are improved by a factor of 24–47% on average for large TSP instances having more than 100 cities using COMFO and can even reach 38–58% using different settings. In addition, compared to other algorithms in the literature, the proposed algorithm provides, on average, better solutions. Hence, it can be considered a promising and efficient technique for this type of problems.

Comparing the effectiveness of the genetic algorithm and ant colony optimization algorithms for the traveling salesman problem

In this paper, we apply the genetic algorithm and ant colony optimization algorithms, which is a kind of meta-heuristics search algorithm, for the traveling salesman problem. We perform experiments to evaluate which one among these algorithms solves the problem more efficiently by means of the solution quality and the execution time. The experimental results show that the ant colony optimization algorithms are efficient in terms of the solution quality, while the genetic algorithm is efficient in terms of the execution time for large traveling salesman problems

A Hybrid PSO-GA Algorithm for Traveling Salesman Problems in Different Environments

In this study particle swarm optimization (PSO) is modified and hybridised with genetic algorithm (GA) using one’s output as the other's input to solve Traveling Salesman Problem(TSP). Here multiple velocity update rules are introduced to modify the PSO and at the time of the movement of a solution, one rule is selected depending on its performances using roulette wheel selection process. Each velocity update rule and the corresponding solution update rule are defined using swap sequence (SS) and swap operation (SO). K-Opt operation is applied in a regular interval of iterations for the movement of any stagnant solution. GA is applied on the final output swarm of the PSO to search the optimal path of the large size TSPs. Roulette wheel selection process, multi-point cyclic crossover and the K-opt operation for the mutation are used in the GA phase. The algorithm is tested in crisp environment using different size benchmark test problems available in the TSPLIB. In the crisp environment the algorithm gives approximately 100% success rate for the test problems up to considerably large sizes. Efficiency of the algorithm is tested with some other existing algorithms in the literature using Friedman test. Some approaches are incorporated with this algorithm for finding solutions of the TSPs in imprecise (fuzzy/rough) environment. Imprecise problems are generated from the crisp problems randomly, solved and obtained results are discussed. It is observed that the performance of the proposed algorithm is better compared to the some other algorithms in the existing literature with respect to the accuracy and the consistency for the symmetric TSPs as well as the Asymmetric TSPs.

Speeding up the patch ordering method for image denoising

Smooth ordering of local patches (patch ordering) has been shown to give state-of-the-art results for image denoising. However, use of very large TSPs (Traveling Salesman Problem) makes it computationally intensive. The patch ordering method forms two large TSPs and employs their approximate solutions in a filtering process to perform denoising. On average, 84% of patch ordering’s execution time was found to be spent on solving TSPs. A variation of the patch ordering method is proposed with two changes. First, numerous smaller TSPs are formed instead of two large ones. Second, the filtering process is modified to perform denoising with solutions of numerous smaller TSPs instead of two large TSPs. Compared to the patch ordering method, the proposed method can denoise images 3.34 times faster. In terms of PSNR, the proposed method’s denoising performance differed by only 0.06 dB on average. Moreover, the proposed method is highly amenable to parallelization. By solving TSPs in parallel, the proposed method’s parallel implementation denoised images 4.89 times faster using four CPU cores which reduced denoising time by 80%. Also shown is that given the same computing resources (CPU cores), the proposed method shall attain speedups higher than those by a similarly parallelized version of the patch ordering method. The proposed approach can be used to speed up patch ordering for other image processing tasks.

A new hybrid gravitational particle swarm optimisation-ACO with local search mechanism, PSOGSA-ACO-Ls for TSP

The travelling salesman problem (TSP) is a hard combinatorial optimisation problem and a popular benchmarking problem at the same time. The TSP has also a number of practical real-world and industrial applications, such as routing in internet of things, IoT, networks, path planning in robotics and many others. In this paper, a new hybrid algorithm for the TSP is proposed; it combines gravitational particle swarm optimisation (PSOGSA) and ACO, and is called ant supervised by gravitational particle swarm optimisation with a local search, PSOGSA-ACO-LS. PSOGSA is used to optimise ACO settings while a local search mechanism, 2-Opt is employed by ACO to ameliorate its local solutions. The proposed method is evaluated using a set a test benches from the TSPLib database including: eil51, berlin52, st70, eil76, rat99, eil101, kroA100, and kroA200. Experimental results show that ACO-GPSO-LS is able to solve the set of TSP instances listed below including the large TSP data sets: kroA100, eli101 and kroA200.

A new hybrid gravitational particle swarm optimisation-ACO with local search mechanism, PSOGSA-ACO-Ls for TSP

The travelling salesman problem (TSP) is a hard combinatorial optimisation problem and a popular benchmarking problem at the same time. The TSP has also a number of practical real-world and industrial applications, such as routing in internet of things, IoT, networks, path planning in robotics and many others. In this paper, a new hybrid algorithm for the TSP is proposed; it combines gravitational particle swarm optimisation (PSOGSA) and ACO, and is called ant supervised by gravitational particle swarm optimisation with a local search, PSOGSA-ACO-LS. PSOGSA is used to optimise ACO settings while a local search mechanism, 2-Opt is employed by ACO to ameliorate its local solutions. The proposed method is evaluated using a set a test benches from the TSPLib database including: eil51, berlin52, st70, eil76, rat99, eil101, kroA100, and kroA200. Experimental results show that ACO-GPSO-LS is able to solve the set of TSP instances listed below including the large TSP data sets: kroA100, eli101 and kroA200.

POPMUSIC for the travelling salesman problem

POPMUSIC— Partial OPtimization Metaheuristic Under Special Intensification Conditions — is a template for tackling large problem instances. This metaheuristic has been shown to be very efficient for various hard combinatorial problems such as p-median, sum of squares clustering, vehicle routing, map labelling and location routing. A key point for treating large Travelling Salesman Problem (TSP) instances is to consider only a subset of edges connecting the cities. The main goal of this article is to present how to build a list of good candidate edges with a complexity lower than quadratic in the context of TSP instances given by a general function. The candidate edges are found with a technique exploiting tour merging and the POPMUSIC metaheuristic. When these candidate edges are provided to a good local search engine, high quality solutions can be found quite efficiently. The method is tested on TSP instances of up to several million cities with different structures (Euclidean uniform, clustered, 2D to 5D, grids, toroidal distances). Numerical results show that solutions of excellent quality can be obtained with an empirical complexity lower than quadratic without exploiting the geometrical properties of the instances.

Water Flow-Like Algorithm with Simulated Annealing for Travelling Salesman Problems

Water Flow-like Algorithm (WFA) has been proved its ability obtaining a fast and quality solution for solving Travelling Salesman Problem (TSP). The WFA uses the insertion move with 2-neighbourhood search to get better flow splitting and moving decision. However, the algorithms can be improved by making a good balance between its solution search exploitation and exploration. Such improvement can be achieved by hybridizing good search algorithm with WFA. This paper presents a hybrid of WFA with various three neighbourhood search in Simulated Annealing (SA) for TSP problem. The performance of the proposed method is evaluated using 18 large TSP benchmark datasets. The experimental result shows that the hybrid method has improved the solution quality compare with the basic WFA and state of art algorithm for TSP.

Social structure optimization in team formation

This paper presents a mathematical framework for treating the Team Formation Problem explicitly incorporating Social Structure (TFP-SS), the formulation of which relies on modern social network analysis theories and metrics. While recent research qualitatively establishes the dependence of team performance on team social structure, the presented framework introduces models that quantitatively exploit such dependence. Given a pool of individuals, the TFP-SS objective is to assign them to teams so as to achieve an optimal structure of individual attributes and social relations within the teams. The paper explores TFP-SS instances with measures based on such network structures as edges, full dyads, triplets, k-stars, etc., in undirected and directed networks. For an NP-Hard instance of TFP-SS, an integer program is presented, followed by a powerful Lin–Kernighan-TFP (LK-TFP) heuristic that performs variable-depth neighborhood search. The idea of such λ-opt sequential search was first employed by Lin and Kernighan, and refined by Helsgaun, for successfully treating large Traveling Salesman Problem instances but has seen limited use in other applications. This paper describes LK-TFP as a tree search procedure and discusses the reasons of its effectiveness. Computational results for small, medium and large TFP-SS instances are reported using LK-TFP and compared with those of an exact algorithm (CPLEX) and a Standard Genetic Algorithm (SGA). Finally, the insights generated by the presented framework and directions for future research are discussed.

A hybrid heuristic algorithm for single and multi-objective imprecise traveling salesman problems

Normally, Travelling salesman problems (TSPs) are formulated with deterministic parameters and either total cost or time is minimized. In real life TSPs, the travel costs and times are not defined precisely and represented by fuzzy or rough data. Very often, in addition to single objective cost or time, both total cost and time are also minimized. These uncertain problem are difficult to optimize. In this paper, some TSPs are formulated as linear programming's problems with imprecise data and there cost, time, or both are minimized by a hybrid heuristic algorithm combining Ant colony optimization (ACO) and Genetic algorithm (GA). Here, hybrid algorithm consumes less resources such as CPU time, then the single heuristic methods. Developed algorithm is capable of solving both single and multi-objective constrained large TSPs with crisp, fuzzy and rough data. In the algorithm, different types of crossovers (multi-point crossover, order crossover, partially mapped crossover) and mutations (single point, multi point) are randomly used. Performance of the algorithm is tested against standard test problems from TSPLIB. Proposed TSPs are solved with proposed and existing algorithms and results are compared. Both the problems and algorithm are illustrated with numerical examples. Some sensitivity analyses are also presented.

Solving a Reverse Supply Chain TSP by Genetic Algorithm <sup></sup>

Stringent legislative, social concerns & clean carbon emissions are constraining companies to take a fresh look at the impact of supply chain operations on environment, society and individuals when designing reverse supply chain networks. A challenging task in today’s globalised environment where companies mandatorily have to collect back goods after its reliable life is making companies integrate supply chain decisions objective here is to minimize transportation cost and distance. In this paper problem of designing a TSP (Travelling Sales man problem) is addressed which is one of the NP-hard problem in combinatorial optimization. Computational experiments conducted with GA (Genetic algorithm) on large and small size TSP cases where compared with NNA (Nearest neighboring algorithm) and have proved that the GA provides optimal tour every time in reasonable time by outperforming the NNA solution when number of cities are increased.

AN IMPROVED ARTIFICIAL BEE COLONY ALGORITHM WITH LOCAL SEARCH FOR TRAVELING SALESMAN PROBLEM

This study aims to solve the traveling salesman problem for small, medium, and large traveling salesman problems taken from the TSPLIB with known solutions, by using an improved artificial bee colony algorithm that is a swarm intelligence-based heuristic algorithm. The improvement process is achieved by using a loyalty function that is used in bee colony optimization as a fitness function used in ABC algorithms. Obtained solutions are compared to solutions from the TSPLIB and the results of Çunkaş and Özsağlam's (2009) study, which includes the solutions for benchmark problems and cities and counties in Turkey according to the genetic algorithm and particle swarm optimization.