Traveling Salesman Problem Instances Research Articles

Ant colony optimization for traveling salesman problem based on parameters optimization

Traveling salesman problem (TSP) is one typical combinatorial optimization problem. Ant colony optimization (ACO) is useful for solving discrete optimization problems whereas the performance of ACO depends on the values of parameters. The hybrid symbiotic organisms search (SOS) and ACO algorithm (SOS–ACO) is proposed for TSP. After certain parameters of ACO are assigned, the remaining parameters can be adaptively optimized by SOS. Using the optimized parameters, ACO finds the optimal or near-optimal solution and the complexity for assigning ACO parameters is greatly reduced. In addition, one simple local optimization strategy is incorporated into SOS–ACO for improving the convergence rate and solution quality. SOS–ACO is tested with different TSP instances in TSPLIB. The best solutions are within 2.33% of the known optimal solution. Compared with some of the previous algorithms, SOS–ACO finds the better solutions under the same preconditions. Finally, the performance of SOS–ACO is analyzed according to the changes of some ACO parameters. The experimental results illustrate that SOS–ACO has good adaptive ability to various values of these parameters for finding the competitive solutions.

Travelling Santa Problem: Optimization of a Million-Households Tour Within One Hour.

Finding the shortest tour visiting all given points at least ones belongs to the most famous optimization problems until today [travelling salesman problem (TSP)]. Optimal solutions exist for many problems up to several ten thousand points. The major difficulty in solving larger problems is the required computational complexity. This shifts the research from finding the optimum with no time limitation to approaches that find good but sub-optimal solutions in pre-defined limited time. This paper proposes a new approach for two-dimensional symmetric problems with more than a million coordinates that is able to create good initial tours within few minutes. It is based on a hierarchical clustering strategy and supports parallel processing. In addition, a method is proposed that can correct unfavorable paths with moderate computational complexity. The new approach is superior to state-of-the-art methods when applied to TSP instances with non-uniformly distributed coordinates.

Semidefinite Programming Relaxations of the Traveling Salesman Problem and Their Integrality Gaps

The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, for example, algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [de Klerk E, Sotirov R (2012) Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Math. Programming 133(1):75–91.] present a semidefinite program (SDP) based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program but generally dominates it on small instances. We provide a family of simplicial TSP instances that shows that the integrality gap of this SDP is unbounded. Second, we show that these simplicial TSP instances imply the unbounded integrality gap of every SDP relaxation of the TSP mentioned in the survey on SDP relaxations of the TSP in section 2 of Sotirov [Sotirov R (2012) SDP relaxations for some combinatorial optimization problems. Anjos MF, Lasserre JB, eds., Handbook on Semidefinite, Conic and Polynomial Optimization (Springer, New York), 795–819.]. In contrast, the subtour linear program performs perfectly on simplicial instances. The simplicial instances thus form a natural litmus test for future SDP relaxations of the TSP.

Discrete Grey Wolf Optimizer for symmetric travelling salesman problem

Grey Wolf Optimizer (GWO) is a recently developed population-based metaheuristic algorithm which imitates the behaviour of grey wolves for survival. Initially, GWO was proposed to solve continuous optimization problems where it performed well. In recent years, numerous versions of GWO are available in the literature and GWO has been widely used to solve engineering problems. This paper presents a novel discrete GWO algorithm (D-GWO) to solve complex discrete travelling salesman problem (TSP). The 2-opt algorithm is incorporated into it to improve the performance of the proposed algorithm. To inspect the performance of D-GWO, the results of the proposed algorithm are compared with well-known metaheuristic algorithms such as Bat Algorithm(BA), Discrete Firefly Algorithm, Imperialist Competitive Algorithm and some other classical algorithms over several known TSP instances. In order to obtain unbiased and rigorous comparison, descriptive statistics such as mean and standard deviation are used as well as statistical tests such as Friedman test and Holm’s test are also conducted. The D-GWO is implemented in MATLAB environment. The computational result carried out in this study has shown that D-GWO outperforms significantly over other alternative algorithms.

Comparison of eleven measures for estimating difficulty of open-loop TSP instances

<abstract> <p>From the theory of algorithms, we know that the time complexity of finding the optimal solution for a traveling salesman problem (TSP) grows exponentially with the number of targets. However, the size of the problem instance is not the only factor that affects its difficulty. In this paper, we review existing measures to estimate the difficulty of a problem instance. We also introduce MST branches and two other measures called greedy path and greedy gap. The idea of MST branches is to generate minimum spanning tree (MST) and then calculate the number of branches in the tree. A branch is a target, which is connected to at least two other targets. We perform an extensive comparison of 11 measures to see how well they correlate to human and computer performance. We evaluate the measures based on time complexity, prediction capability, suitability, and practicality. The results show that while the MST branches measure is simple, fast to compute, and does not need to have the optimal solution as a reference unlike many other measures. It correlates equally good or even better than the best of the previous measures ‑ the number of targets, and the number of targets on the convex hull.</p> </abstract>

On Randomness and Structure in Euclidean TSP Instances: A Study With Heuristic Methods

Prediction of the quality of the result provided by a specific solving method is an important factor when choosing how to solve a given problem. The more accurate the prediction, the more appropriate the decision on what to choose when several solving applications are available. In this article, we study the impact of the structure of a Traveling Salesman Problem instance on the quality of the solution when using two representative heuristics: the population-based Ant Colony Optimization (ACO) and the local search Lin-Kernighan (LK) algorithm. The quality of the result for a solving method is measured by the computation accuracy, which is expressed using the percent error between its solution and the optimum one. We classify the instances in structured, semi-structured, and unstructured and perform a between classes and inside-classes analysis. All the structured instances were solved to optimality by the ACO implementation, which was not the case for the LK application. On small random instances, the ACO implementation used in this paper also optimally found the solutions. We show that the quality of the results on semi-structured and unstructured instances can be predicted using some instance parameters when using the ACO implementation. Using the same parameters, the accuracy of the solutions provided by the Lin-Kernighan application cannot be predicted. We also propose several new structured, clustered, and unstructured 2D Euclidean Traveling Salesman Problem instances that can be used by the research community for further investigations.

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).

Modified ant colony optimization with improved tour construction and pheromone updating strategies for traveling salesman problem

To improve ant colony optimization (ACO) for traveling salesman problem (TSP), its two main strategies which are tour construction and pheromone updating have been modified, and one modified ACO (MACO) has been proposed. For the first strategy (tour construction strategy), one new method to construct tours by combining paths of two meeting ants has been applied. And for second strategy (pheromone updating strategy), one new method to polarize pheromone density of all paths has been proposed. Based on the applications for 40 standard benchmark TSP instances (datasets) ranging from 29 to 13,509 cities, the good performance of the MACO is verified. To verify the MACO deeply, based on the applications for some standard TSP instances, the computing results of MACO are compared with three typical state-of-the-art algorithms based on ACO methods. Moreover, based on the applications for some standard TSP instances, the computing results of MACO are compared with 10 state-of-the-art metaheuristic algorithms. The comparison studies show that the MACO can attain the optimal solution with higher accuracy, no matter how complicated the TSPs are. And its performance is the best, comparing to all state-of-the-art algorithms. At last, two new strategies used in MACO have been analyzed comprehensively by the applications for 25 TSPs. The results show that the first strategy is mainly used to improve the computing efficiency, and the second one is mainly used to improve the computing effect.

Opposition-Based Ant Colony Optimization Algorithm for the Traveling Salesman Problem

Opposition-based learning (OBL) has been widely used to improve many swarm intelligent optimization (SI) algorithms for continuous problems during the past few decades. When the SI optimization algorithms apply OBL to solve discrete problems, the construction and utilization of the opposite solution is the key issue. Ant colony optimization (ACO) generally used to solve combinatorial optimization problems is a kind of classical SI optimization algorithm. Opposition-based ACO which is combined in OBL is proposed to solve the symmetric traveling salesman problem (TSP) in this paper. Two strategies for constructing opposite path by OBL based on solution characteristics of TSP are also proposed. Then, in order to use information of opposite path to improve the performance of ACO, three different strategies, direction, indirection, and random methods, mentioned for pheromone update rules are discussed individually. According to the construction of the inverse solution and the way of using it in pheromone updating, three kinds of improved ant colony algorithms are proposed. To verify the feasibility and effectiveness of strategies, two kinds of ACO algorithms are employed to solve TSP instances. The results demonstrate that the performance of opposition-based ACO is better than that of ACO without OBL.

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.

Multi-core-, multi-thread-based optimization algorithm for large-scale traveling salesman problem

With the rapid development of general hardware technology, microcomputers with multi-core CPUs have been widely applied in commercial services and household usage in the last ten years. Multi-core chips could, theoretically, lead to much better performance and computational efficiency than single-core chips. But so far, they have not shown general advantages for users, other than for operating systems and some specialized software. It is not easy to transform traditional single-core-based algorithms into multi-core-, multi-thread-based algorithms that can greatly improve efficiency, because of difficulties in computation and scheduling of hardware kernels, and because some programming languages cannot support multi-core, multi-thread programming. Therefore, a kind of multi-core-, multi-thread-based fast algorithm was designed and coded with Delphi language for the medium- and large-scale traveling salesman problem instances from TSPLIB, which can fully speed up the searching process without loss of quality. Experimental results show that the algorithm proposed can, under the given hardware limitations, take full advantage of multi-core chips and effectively balance the conflict between increasing problem size and computational efficiency and thus acquire satisfactory solutions.

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.

DEACO: Adopting dynamic evaporation strategy to enhance ACO algorithm for the traveling salesman problem

Ant Colony Optimization (ACO) algorithm is one of the effective solutions to solve the problem of combination optimization like traveling salesman problem (TSP) which belongs to NP-hard problem. However, this algorithm is robust and has a strong ability for solution discovery, but the convergence speed of that is low and stuck into local optimum. Therefore, for overcoming the drawbacks of ACO, we proposed a self-adaptive ACO with unique strategies to improve uncertain convergence time and random decisions of this algorithm. The proposed technique (DEACO) adjusting the ACO parameters dynamically. In this mechanism, main idea is how to select the first city (start point) to achieve the shortest path based on clustering. In this approach, DEACO finds the minimum cost/shortest path for each cluster. The data that used for this experiment is from TSPLIB library under MATLAB simulation with 10 TSP instances. The experiment outcome illustrates better performance of the proposed method than the conventional ACO in term of faster convergence speed and higher search accuracy.

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 $\frac {N*(N-1)}{2}$ 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.

Slope-to-optimal-solution-based evaluation of the hardness of travelling salesman problem instances

AbstractThe travelling salesman problem is one of the most popular problems in combinatorial optimization. It has been frequently used as a benchmark of the performance of evolutionary algorithms. For this reason, nowadays practitioners request new and more difficult instances of this problem. This leads to investigate how to evaluate the intrinsic difficulty of the instances and how to separate ease and difficult instances. By developing methodologies for separating easy- from difficult-to-solve instances, researchers can fairly test the performance of their combinatorial optimizers. In this work, a methodology for evaluating the difficulty of instances of the travelling salesman problem near the optimal solution is proposed. The question is if the fitness landscape near the optimal solution encodes enough information to separate instances in function of their intrinsic difficulty. This methodology is based on the use of a random walk to explore the closeness of the optimal solution. The optimal solution is modified by altering one connection between two cities at each step, at the same time that the fitness of the altered solution is evaluated. This permits evaluating the slope of the fitness landscape. Later, and using the previous information, the difficulty of the instance is evaluated with random forests and artificial neural networks. In this work, this methodology is confronted with a wide set of instances. As a consequence, a methodology to separate the instances of the travelling salesman problem by their degree of difficulty is proposed and evaluated.

Implementing a GPU-based parallel MAX–MIN Ant System

The MAX–MIN Ant System (MMAS) is one of the best-known Ant Colony Optimization (ACO) algorithms proven to be efficient at finding satisfactory solutions to many difficult combinatorial optimization problems. The slow-down in Moore’s law, and the availability of graphics processing units (GPUs) capable of conducting general-purpose computations at high speed, has sparked considerable research efforts into the development of GPU-based ACO implementations. In this paper, we discuss a range of novel ideas for improving the GPU-based parallel MMAS implementation, allowing it to better utilize the computing power offered by two subsequent Nvidia GPU architectures. Specifically, based on the weighted reservoir sampling algorithm we propose a novel parallel implementation of the node selection procedure, which is at the heart of the MMAS and other ACO algorithms. We also present a memory-efficient implementation of another key-component – the tabu list structure – which is used in the ACO’s solution construction stage. The proposed implementations, combined with the existing approaches, lead to a total of six MMAS variants, which are evaluated on a set of Traveling Salesman Problem (TSP) instances ranging from 198 to 3795 cities. The results show that our MMAS implementation is competitive with state-of-the-art GPU-based and multi-core CPU-based parallel ACO implementations: in fact, the times obtained for the Nvidia V100 Volta GPU were up to 7.18x and 21.79x smaller, respectively. The fastest of the proposed MMAS variants is able to generate over 1 million candidate solutions per second when solving a 1002-city instance. Moreover, we show that, combined with the 2-opt local search heuristic, the proposed parallel MMAS finds high-quality solutions for the TSP instances with up to 18,512 nodes.

Cantor set based neighbor generation method for permutation solution representation

Metaheuristics gained world-wide popularity and researchers have been studying them vigorously in the last two decades. A relatively less explored approach in the improvement of metaheuristics is to design new neighbor generation mechanisms. Neighbor generation mechanisms are very important in the success of any single solution-based heuristic since they directly guide the search. In this study, a neighbor generation mechanism called cantor-set based (CB) method for single solution-based heuristics which use permutation solution representation is described. The inspiration for CB method stems from the recursive algorithm that constructs a cantor set which is a fractal set. Three variations of CB method are discussed (CB-1, CB-2, CB-3) considering the presented design possibilities. The computational experiments are conducted by embedding the mechanisms into the classical local search and simulated annealing algorithms, separately, to test their efficiency and effectiveness by comparing them to classical swap and insertion mechanisms. The traveling salesman problem (TSP) and quadratic assignment problem (QAP) which are very different problems that have incompatible characteristics have been chosen to test the mechanisms and sets of benchmark instances with varying sizes are chosen for the comparisons. The computational tests show that CB-2 gives very favorable results for TSP and CB-1 gives favorable results for QAP which means that CB-2 is suitable for problems that have steep landscapes and CB-1 is suitable for the problems that have flat landscapes. It is observed that CB-3 is a more generalized mechanism because it gives consistently good results for both TSP and QAP instances. The best mechanism for a given instance of the both problem types outperforms the classical neighbor generation of swap and insertion in terms of effectiveness.

The salesman’s improved tours for fundamental classes

Finding the exact integrality gap $$\alpha $$ for the LP relaxation of the metric travelling salesman problem (TSP) has been an open problem for over 30 years, with little progress made. It is known that $$4/3 \le \alpha \le 3/2$$ , and a famous conjecture states $$\alpha = 4/3$$ . It has also been conjectured that the integrality gap is achieved for half-integer basic solutions of the linear program. For this problem, essentially two “fundamental” classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most $$\rho $$ for all instances of the metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the 1 / 2-integer points of one such class. We successfully improve the upper bound for the integrality gap from 3 / 2 to 10 / 7 for a superclass of these points for which a lower bound of 4 / 3 is proved. A key role in the proof of this result is played by finding Hamiltonian cycles whose existence is equivalent to Kotzig’s result on “compatible Eulerian tours”, and which lead us to delta-matroids for developing the related algorithms. Our arguments also involve other innovative tools from combinatorial optimization with the potential of a broader use.

Discrete Spider Monkey Optimization for Travelling Salesman Problem

Meta-heuristic algorithms inspired by biological species have become very popular in recent years. Collective intelligence of various social insects such as ants, bees, wasps, termites, birds, fish, has been investigated to develop a number of meta-heuristic algorithms in the general domain of swarm intelligence (SI). The developed SI algorithms are found effective in solving different optimization tasks. Travelling Salesman Problem (TSP) is the combinatorial optimization problem where a salesman starting from a home city travels all the other cities and returns to home city in the shortest possible path. TSP is a popular problem due to the fact that the instances of TSP can be applied to solve real-world problems, implication of which turns TSP into a standard test bench for performance evaluation of new algorithms. Spider Monkey Optimization (SMO) is a recent addition to SI algorithms based on the social behaviour of spider monkeys. SMO implicitly adopts grouping and regrouping for the interactions to improve solutions; such multi-population approach is the motivation of this study to develop an effective method for TSP. This paper presents an effective variant of SMO to solve TSP called discrete SMO (DSMO). In DSMO, every spider monkey represents a TSP solution where Swap Sequence (SS) and Swap Operator (SO) based operations are employed, which enables interaction among monkeys in obtaining the optimal TSP solution. The SOs are generated using the experience of a specific spider monkey as well as the experience of other members (local leader, global leader, or a randomly selected spider monkey) of the group. The performance and effectiveness of the proposed method have been verified on a large set of TSP instances and the outcomes are compared to other well-known methods. Experimental results demonstrate the effectiveness of the proposed DSMO for solving TSP.

The multicommodity traveling salesman problem with priority prizes: a mathematical model and metaheuristics

The classic Traveling Salesman Problem (TSP) only considers the costs involved in the routes and does not differentiate products or customers. Logistic companies face conflict between operational costs, customers with different categories of products, and customer satisfaction, which is directly related to delivery time. This paper presents a new mathematical model for a TSP with variable costs and priority prizes, taking into account the customer’s product and preference values. This problem is denoted as the Multicommodity Traveling Salesman Problem with Priority Prizes (MTSPPP). Two versions of the Biased Random-Key Genetic Algorithm (BRKGA) are proposed to solve medium and large instances of the MTSPPP. Computational tests were performed, using modified instances based on classical TSP instances. The proposed methods have proved to be efficient in solving the MTSPPP.