Symmetric Traveling Salesman Problem Research Articles

Framework for Small Traveling Salesman Problems

We study small traveling salesman problems (TSPs) because current quantum computers can find optional solutions for TSPs with up to 14 cities. Also, we study small TSPs because TSPs have been recommended to be benchmarks to measure quantum optimization on all types of quantum hardware. This means comparisons of quantum data about small TSPs. We extent previous numerical results that were reported in “Small Traveling Salesman Problems” for 6, 8 and 10 cities. The new results in this paper are for 10 – 14 cities in symmetric TSPs. The data for this new range of cities is consistent with the previous data and can be the basis for estimates of results from quantum computers that are upgraded to handle more than 14 cities. The work and analysis suggest two conjectures that we discuss. The paper also contains an annotated survey of recent publications about TSPs.

Branch and Price for the Stochastic Traveling Salesman Problem with Generalized Latency

We consider an extension of the symmetric traveling salesman problem with generalized latency that explicitly models uncertainty. The stochastic traveling salesman problem with generalized latency (STSP-GL) aims to choose a subset of nodes of an undirected graph and determines a Hamiltonian tour among those nodes, minimizing an objective function that is a weighted combination of route design and passenger routing costs. These nodes are selected to ensure that a predefined percentage of uncertain passenger demand is served with a given probability. We formulate the STSP-GL as a stochastic program and propose a branch-and-price algorithm for solving its deterministic equivalent. We also develop a local search approach with which we improve the performance of the branch-and-price approach. We assess the efficiency of the proposed methods on a set of instances from the literature. We demonstrate that the proposed methods outperform a known benchmark, improving upper bounds by up to 85% and lower bounds by up to 55%. Finally, we show that solutions of the stochastic model are both more cost-effective and robust than those of the deterministic model. History: This paper has been accepted for the Transportation Science Special Issue on TSL Conference 2023. Funding: This work was supported by the Bundesministerium für Bildung und Forschung [Grant 03ZU1105FA]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0417 .

AutoRL-Sim: Automated Reinforcement Learning Simulator for Combinatorial Optimization Problems

Reinforcement learning is a crucial area of machine learning, with a wide range of applications. To conduct experiments in this research field, it is necessary to define the algorithms and parameters to be applied. However, this task can be complex because of the variety of possible configurations. In this sense, the adoption of AutoRL systems can automate the selection of these configurations, simplifying the experimental process. In this context, this work aims to propose a simulation environment for combinatorial optimization problems using AutoRL. The AutoRL-Sim includes several experimentation modules that cover studies on the symmetric traveling salesman problem, the asymmetric traveling salesman problem, and the sequential ordering problem. Furthermore, parameter optimization is performed using response surface models. The AutoRL-Sim simulator allows users to conduct experiments in a more practical way, without the need to worry about implementation. Additionally, they have the ability to analyze post-experiment data or save them for future analysis.

Two-Stage Probe-Based Search Optimization Algorithm for the Traveling Salesman Problems

As a classical combinatorial optimization problem, the traveling salesman problem (TSP) has been extensively investigated in the fields of Artificial Intelligence and Operations Research. Due to being NP-complete, it is still rather challenging to solve both effectively and efficiently. Because of its high theoretical significance and wide practical applications, great effort has been undertaken to solve it from the point of view of intelligent search. In this paper, we propose a two-stage probe-based search optimization algorithm for solving both symmetric and asymmetric TSPs through the stages of route development and a self-escape mechanism. Specifically, in the first stage, a reasonable proportion threshold filter of potential basis probes or partial routes is set up at each step during the complete route development process. In this way, the poor basis probes with longer routes are filtered out automatically. Moreover, four local augmentation operators are further employed to improve these potential basis probes at each step. In the second stage, a self-escape mechanism or operation is further implemented on the obtained complete routes to prevent the probe-based search from being trapped in a locally optimal solution. The experimental results on a collection of benchmark TSP datasets demonstrate that our proposed algorithm is more effective than other state-of-the-art optimization algorithms. In fact, it achieves the best-known TSP benchmark solutions in many datasets, while, in certain cases, it even generates solutions that are better than the best-known TSP benchmark solutions.

Improvement of the Nearest Neighbor Heuristic Search Algorithm for Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is classified as a non-deterministic polynomial (NP) hard problem, which has found widespread application in several scientific and technological domains. Due to its NP-hard nature, it is very hard to solve effectively and efficiently. Despite this rationale, a multitude of optimization approaches have been proposed and developed by scientists and researchers during the last several decades. Among these several algorithms, heuristic approaches are deemed appropriate for addressing this intricate issue. One of the simplest and most easily implementable heuristic algorithms for TSP is the nearest neighbor algorithm (NNA). However, its solution quality suffers owing to randomness in the optimization process. To address this issue, this study proposes a deterministic NNA for solving symmetric TSP. It is an improved version of NNA, which starts with the shortest edge consisting of two cities and then repeatedly includes the closest city on the route until an effective route is established. The simulation is conducted on 20 benchmark symmetric TSP datasets obtained from TSPLIB. The simulation results provide evidence that the improved NNA outperforms the basic NNA throughout most of the datasets in terms of solution quality as well as computational time.

A New Approach to Solve the Classical Symmetric Traveling Salesman Problem by Highest Suffix Method

A New Approach to Solve the Classical Symmetric Traveling Salesman Problem by Highest Suffix Method

Loop-Wise Route Representation and Its Optimization Formulation for Symmetric Traveling Salesman Problems

This study proposes a new loop-wise route representation in which a vehicle route is represented in terms of continuous loop variables and a base route. The proposed loop-wise representation can guarantee a local route connectivity at each node with no constraint function, thereby simplifying the optimization formulation. Based on the proposed loop-wise route representation, optimization formulation is expressed for the traveling salesman problems (TSPs) in the planar graph. The performance of the proposed method is compared with that of genetic algorithm (GA) methods in four different cases of the TSPs. Numerical results show that the proposed method can determine the optimal route for the TSPs with a much higher computational efficiency, compared with the conventional GA methods.

Review of Swarm Intelligence for Solving Symmetric Traveling Salesman Problem

Swarm Intelligence algorithms are computational intelligence algorithms inspired from the collective behavior of real swarms such as ant colony, fish school, bee colony, bat swarm, and other swarms in the nature. Swarm Intelligence algorithms are used to obtain the optimal solution for NP-Hard problems that are strongly believed that their optimal solution cannot be found in an optimal bounded time. Travels Salesman Problem (TSP) is an NP-Hard problem in which a salesman wants to visit all cities and return to the start city in an optimal time. In this article we are applying most efficient heuristic based Swarm Intelligence algorithms which are Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), Bat algorithm (BA), and Ant Colony Optimization (ACO) algorithm to find a best solution for TSP which is one of the most well-known NP-Hard problems in computational optimization. Results are given for different TSP problems comparing the best tours founds by BA, ABC, PSO and ACO.

An Improvement to the 2-Opt Heuristic Algorithm for Approximation of Optimal TSP Tour

The travelling salesman problem (TSP) is perhaps the most researched problem in the field of Computer Science and Operations. It is a known NP-hard problem and has significant practical applications in a variety of areas, such as logistics, planning, and scheduling. Route optimisation not only improves the overall profitability of a logistic centre but also reduces greenhouse gas emissions by minimising the distance travelled. In this article, we propose a simple and improved heuristic algorithm named 2-Opt++, which solves symmetric TSP problems using an enhanced 2-Opt local search technique, to generate better results. As with 2-Opt, our proposed method can also be applied to the Vehicle Routing Problem (VRP), with minor modifications. We have compared our technique with six existing algorithms, namely ruin and recreate, nearest neighbour, genetic algorithm, simulated annealing, Tabu search, and ant colony optimisation. Furthermore, to allow for the complexity of larger TSP instances, we have used a graph compression/candidate list technique that helps in reducing the computational complexity and time. The comprehensive empirical evaluation carried out for this research work shows the efficacy of the 2-Opt++ algorithm as it outperforms the other well-known algorithms in terms of the error margin, execution time, and time of convergence.

Adjustable Pheromone Reinforcement Strategies for Problems with Efficient Heuristic Information

Ant colony optimization (ACO) is a well-known class of swarm intelligence algorithms suitable for solving many NP-hard problems. An important component of such algorithms is a record of pheromone trails that reflect colonies’ experiences with previously constructed solutions of the problem instance that is being solved. By using pheromones, the algorithm builds a probabilistic model that is exploited for constructing new and, hopefully, better solutions. Traditionally, there are two different strategies for updating pheromone trails. The best-so-far strategy (global best) is rather greedy and can cause a too-fast convergence of the algorithm toward some suboptimal solutions. The other strategy is named iteration best and it promotes exploration and slower convergence, which is sometimes too slow and lacks focus. To allow better adaptability of ant colony optimization algorithms we use κ-best, max-κ-best, and 1/λ-best strategies that form the entire spectrum of strategies between best-so-far and iteration best and go beyond. Selecting a suitable strategy depends on the type of problem, parameters, heuristic information, and conditions in which the ACO is used. In this research, we use two representative combinatorial NP-hard problems, the symmetric traveling salesman problem (TSP) and the asymmetric traveling salesman problem (ATSP), for which very effective heuristic information is widely known, to empirically analyze the influence of strategies on the algorithmic performance. The experiments are carried out on 45 TSP and 47 ATSP instances by using the MAX-MIN ant system variant of ACO with and without local optimizations, with each problem instance repeated 101 times for 24 different pheromone reinforcement strategies. The results show that, by using adjustable pheromone reinforcement strategies, the MMAS outperformed in a large majority of cases the MMAS with classical strategies.

The Solutions to Traveling Salesman Problem

This paper presents solutions to symmetric, asymmetric, and multiple traveling salesman problems. In the symmetric traveling salesman problem, one salesperson must go to five cities, making precisely one stop at each location. A matrix with the distances between each city is provided. Using the branch and bound algorithm and expressing it in Python, the final result is obtained. The formulations of the asymmetric traveling salesman problem and the multiple traveling salesman problem are demonstrated in the paper. The asymmetric problem in the paper is solved by transforming the asymmetric traveling salesman problem into a symmetric traveling salesman problem; then, the branch and bound algorithm has been applied to solve the problem. In the multiple traveling salesman problem, three salesmen are required to visit seven cities in total, each of which is only visited by one salesperson once. Each city is a point in an x-y plane, and the coordinates of all points are given in a graph. By formulating the MTSP problem with an assignment-based double-index integer programming and applying the constraints, the outcome is derived from Python code within a few seconds.

A Multigoal Path-Planning Approach for Explosive Ordnance Disposal Robots Based on Bidirectional Dynamic Weighted-A* and Learn Memory-Swap Sequence PSO Algorithm

In order to protect people’s lives and property, increasing numbers of explosive disposal robots have been developed. It is necessary for an explosive ordinance disposal (EOD) robot to quickly detect all explosives, especially when the location of the explosives is unknown. To achieve this goal, we propose a bidirectional dynamic weighted-A star (BD-A*) algorithm and a learn memory-swap sequence particle swarm optimization (LM-SSPSO) algorithm. Firstly, in the BD-A* algorithm, our aim is to obtain the shortest distance path between any two goal positions, considering computation time optimization. We optimize the computation time by introducing a bidirectional search and a dynamic OpenList cost weight strategy. Secondly, the search-adjacent nodes are extended to obtain a shorter path. Thirdly, by using the LM-SSPSO algorithm, we aim to plan the shortest distance path that traverses all goal positions. The problem is similar to the symmetric traveling salesman problem (TSP). We introduce the swap sequence strategy into the traditional PSO and optimize the whole PSO process by imitating human learning and memory behaviors. Fourthly, to verify the performance of the proposed algorithm, we begin by comparing the improved A* with traditional A* over different resolutions, weight coefficients, and nodes. The hybrid PSO algorithm is also compared with other intelligent algorithms. Finally, different environment maps are also discussed to further verify the performance of the algorithm. The simulation results demonstrate that our improved A* algorithm has superior performance by finding the shortest distance with less computational time. In the simulation results for LM-SSPSO, the convergence rate significantly improves, and the improved algorithm is more likely to obtain the optimal path.

Simulated Annealing – 2 Opt Algorithm for Solving Traveling Salesman Problem

The purpose of this article is to elaborate performance of the hybrid model of Simulated Annealing (SA) and 2 Opt algorithm for solving the traveling salesman problem (TSP). The SA algorithm used in this article is based on the outer and inner loop SA algorithm. The hybrid algorithm has promising results in solving small and medium-scale symmetric traveling salesman problem benchmark tests taken from the TSPLIB reference. Results of the optimal solution and standard deviation indicate that the hybrid algorithm shows good performance in terms of reliability and stability in finding the optimal solution from the TSP benchmark case. Values of average error and standard deviation for all simulations in the medium scale are 0.0267 and 644.12, respectively. Moreover, in some cases namely KroB100, Pr107, and Pr144, the hybrid algorithm finds a better solution compared with the best-known solution mentioned in the reference. Further, the hybrid algorithm is 1.207 – 5.692 times faster than the pure outer and inner loop-based SA algorithm. Additionally, the results show that the hybrid algorithm outperforms other hybrid algorithms such as SA – nearest neighbor (NN) and NN – 2 Opt.

Discrete Salp Swarm Algorithm for symmetric traveling salesman problem.

In the Salp Swarm Algorithm (SSA), the update mechanism is inspired by the unique chain movement of the salp swarm. Numerous versions of SSA were already put forward to deal with various optimization problems, but there are very few discrete versions among them. d-opt is improved based on the 2-opt algorithm: a decreasing factor d is introduced to control the range of neighborhood search; TPALS are modified by Problem Aware Local Search (PALS) based on the characteristics of Travelling Salesman Problem (TSP); The second leader mechanism increases the randomness of the algorithm and avoids falling into the local optimal solution to a certain extent. We also select six classical crossover operators to experiment and select Subtour Exchange Crossover (SEC) and the above three mechanisms to integrate them into the SSA algorithm framework to form Discrete Salp Swarm Algorithm (DSSA). In addition, DSSA was tested on 23 known TSP instances to verify its performance. Comparative simulation studies with other advanced algorithms are conducted and from the results, it is observed that DSSA satisfactorily solves TSP.

The Circlet Inequalities: A New, Circulant-Based, Facet-Defining Inequality for the TSP

Facet-defining inequalities of the symmetric traveling salesman problem (TSP) polytope play a prominent role in both polyhedral TSP research and state-of-the-art TSP solvers. In this paper, we introduce a new class of facet-defining inequalities, the circlet inequalities. These inequalities were first conjectured in Gutekunst and Williamson [Gutekunst SC, Williamson DP (2019) Characterizing the integrality gap of the subtour LP for the circulant traveling salesman problem. SIAM J. Discrete Math. 33(4):2452–2478] when studying the circulant TSP, and they provide a bridge between polyhedral TSP research and number-theoretic investigations of Hamiltonian cycles stemming from a conjecture from Marco Buratti in 2007. The circlet inequalities exhibit circulant symmetry by placing the same weight on all edges of a given length; our main proof exploits this symmetry to prove the validity of the circlet inequalities. We then show that the circlet inequalities are facet-defining and compute their strength following Goemans [Goemans MX (1995) Worst-case comparison of valid inequalities for the TSP. Math. Programming 69:335–349]; they achieve the same worst case strength as the similarly circulant crown inequalities of Naddef and Rinaldi [Naddef D, Rinaldi G (1992) The crown inequalities for the symmetric traveling salesman polytope. Math. Oper. Res. 17(2):308–326] but are generally stronger. Funding: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program [Grant DGE-1650441] and by the National Science Foundation Division of Computing and Communications Foundations [Grant CCF-1908517].

A new integer programming formulation of the graphical traveling salesman problem

In the Traveling Salesman Problem (TSP), a salesman wants to visit a set of cities and return home. There is a cost $$c_{ij}$$ of traveling from city i to city j, which is the same in either direction for the Symmetric TSP. The objective is to visit each city exactly once, minimizing total travel costs. In the Graphical TSP, a city may be visited more than once, which may be necessary on a sparse graph. We present a new integer programming formulation for the Graphical TSP requiring only two classes of polynomial-sized constraints while addressing an open question proposed by Denis Naddef. We generalize one of these classes, and present promising preliminary computational results.

The Ant Heuristic “Reloaded”

As a swarm-based optimization heuristic, the Ant System (AS) was proposed to deal with the Traveling Salesman Problem (TSP). Classifified as a constructive approach, AS was inspired by the ants’ social behavior.In Practice, its implementation reveals three basic variants that exploit two operating models: a fifirst “natural” model, where ants move and update pheromones, with consideration of distances between towns, to defifine the Ant-Quantity variant, and without considering distances for the Ant-Density variant. Or an “abnormal” second model, where ants move and delay updates until all ants have completed a full cycle, then consider the tour length which defifines the Ant-Cycle that was claimed to be the best variant. We reload the AS and reconsider these three basic ant algorithms and their respective two models. We propose to explore an “overlooked” third model, which consists of further expanding the “abnormal” ants’ attitude, forcing them to move and update pheromones, simultaneously, after every single move, thus conceiving the Ant-Step model. Keeping the option whether or not to consider distances, we propose two new AS basic algorithms: the Ant-Step-Quantity and the Ant-Step-Density. The aim of this paper is to present and assess these two new basic AS variants in respect to the three existing ones, specially the one considered the best, through an experimental study on various symmetric TSP benchmarks.

An Improved Unordered Pair Bat Algorithm for Solving the Symmetrical Traveling Salesman Problem

Abstract Bat algorithm is an effective swarm intelligence optimization algorithm which is widely used to solve continuous optimization problems. But it still has some limitations in search process and can’t solve discrete optimization problems directly. Therefore, this paper introduces an unordered pair and proposes an unordered pair bat algorithm (UPBA) to make it more suitable for solving symmetric discrete traveling salesman problems. To verify the effectiveness of this method, the algorithm has been tested on 23 symmetric benchmarks and compared its performance with other algorithms. The results have shown that the proposed UPBA outperforms all the other alternatives significantly in most cases.

Discrete sparrow search algorithm for symmetric traveling salesman problem

The traveling salesman problem (TSP) is one of the most intensively studied problems in computational mathematics. This paper proposes a swarm intelligence approach using a discrete sparrow search algorithm (DSSA) with a global perturbation strategy to solve the problem. Firstly, the initial solution in the population is generated by the roulette-wheel selection. Secondly, the order-based decoding method is introduced to complete the update of the sparrow position. Then, the global perturbation mechanism combined with Gaussian mutation and swap operator is adopted to balance exploration and exploitation capability. Finally, the 2-opt local search is integrated to further improve the quality of the solution. Those strategies enhance the solution’s quality and accelerate the convergence. Experiments on 34 TSP benchmark datasets are conducted to investigate the performance of the proposed DSSA. And statistical tests are used to verify the significant differences between the proposed DSSA and other state-of-the-art methods. Results show that the proposed method is more competitive and robust in solving the TSP.

Are Symmetric Traveling Salesman Problems Well Suited to Benchmark Some Quantum Optimization Problems?

Are Symmetric Traveling Salesman Problems Well Suited to Benchmark Some Quantum Optimization Problems?