Traveling Salesman Problem Instances Research Articles

An iterative algorithm to eliminate edges for traveling salesman problem based on a new binomial distribution

Traveling salesman problem (TSP) is one of the extensively studied NP-hard problems. The recent research showed that the TSP on sparse graphs could be resolved in the relatively shorter computation time than that on the complete graph $K_{n}$ . This paper updates a previous probability model for the optimal Hamiltonian cycle edges according to the frequency quadrilaterals in $K_{n}$ . A new binomial distribution for TSP is rebuilt to show the probability that an edge e has the frequency 5 in a frequency quadrilateral. Based on the binomial distribution, an iterative algorithm is designed to compute the sparse graphs for TSP. There are two steps at each computation cycle. Firstly, N frequency quadrilaterals containing an edge e in the input graph is chosen to compute the average frequency $\bar {f}(e)$ with the frequency quadrilaterals where e has the frequency 5. Secondly, half edges with the small values $\bar {f}(e)$ are eliminated. The two steps are repeated until a sparse graph is computed. The computation time of the algorithm is $O(Nn^{2})$ . For the TSP instances in the TSPLIB, the experimental results illustrated that the sparse graphs with the $O(n\log _{2} n)$ edges are computed and the original optimal solution is preserved. The experiments means the optimal Hamiltonian cycle edges have the bigger average frequency $\bar {f}(e)$ in $K_{n}$ and the subgraphs of $K_{n}$ so they are preserved in the computation process.

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.

Fixed-parameter algorithms for rectilinear Steiner tree and rectilinear traveling salesman problem in the plane

Given a set P of n points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the l1 distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in O(nh7h) where h ≤ n denotes the number of horizontal lines containing the points of P. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of P providing a O(nh5h) time complexity. Both bounds improve over the best time bounds known for these problems.

Combinatorial GVNS (General Variable Neighborhood Search) Optimization for Dynamic Garbage Collection

General variable neighborhood search (GVNS) is a well known and widely used metaheuristic for efficiently solving many NP-hard combinatorial optimization problems. We propose a novel extension of the conventional GVNS. Our approach incorporates ideas and techniques from the field of quantum computation during the shaking phase. The travelling salesman problem (TSP) is a well known NP-hard problem which has broadly been used for modelling many real life routing cases. As a consequence, TSP can be used as a basis for modelling and finding routes via the Global Positioning System (GPS). In this paper, we examine the potential use of this method for the GPS system of garbage trucks. Specifically, we provide a thorough presentation of our method accompanied with extensive computational results. The experimental data accumulated on a plethora of TSP instances, which are shown in a series of figures and tables, allow us to conclude that the novel GVNS algorithm can provide an efficient solution for this type of geographical problem.

Multi-agent learning technique inspired from software engineering model for permutation coded GA

Hybrid genetic algorithms (HGAs) have recognized significant attention in recent years and being progressively more used to solve real-world problems. HGA is the forms of assimilation between genetic algorithm and other search optimization techniques to improve the overall performance of the genetic algorithm (GA). GA can be hybridized with many other bio-inspired heuristic algorithms such as particle swarm optimization, cuckoo search genetic algorithm, firefly algorithm-genetic algorithm, real coded genetic algorithm-artificial fish swarm algorithm, etc., multi-agent system (MAS) is a new paradigm for conceptualizing, designing and optimizing the solution models. Multi-agent learning concept improves the outcome of the MAS and the type of learning process involved in it plays a vital role. Incremental model is a well-known software development process in which the system yields better effect at the end of every cycle. In this paper, software engineering inspired incremental model based multi-agent learning model for permutation coded genetic algorithms has been proposed. One of the famous combinatorial hard problems of traveling salesman problem (TSP) is being chosen as the test bed and the experiments are performed on large sized benchmark TSP instances obtained from standard TSPLIB. The experimental results support that the proposed model perform better than existing best working initialization methods in terms of convergence rate, error rate and computation time.

A quick method to compute sparse graphs for traveling salesman problem using random frequency quadrilaterals

Traveling salesman problem (TSP) is extensively studied in combinatorial optimization and computer science. This paper gives a quick method to compute the sparse graphs for TSP based on the random frequency quadrilaterals so as to reduce the TSP on the complete graph to the TSP on the sparse graphs. When we choose N frequency quadrilaterals containing an edge e to compute its total frequency, the frequency of e in the optimal Hamiltonian cycle will be bigger than that of most of the other edges. We fix N to compute the frequency of each edge and the computation time of the quick method is O(n2). We suggest two frequency thresholds to trim the edges with the frequency below the two frequency thresholds and generate the sparse graphs for TSP. The experimental results show we compute the sparse graphs for these TSP instances in the TSPLIB.

Solving Travelling Salesman Problem Using Heart Algorithm

Solving hard problems like the Travelling Salesman Problem (TSP) is a major challenge faced by analysts even though many techniques are available. The main goal of TSP is that a number of cities should be visited by a salesman and return to the starting city along a number of possible shortest paths. TSP is although looking a simple problem, but it is an important problem of the classical optimization problems that are difficult to solve conventionally. It has been proved that solving TSP by the conventional approaches in a reasonable time is not possible. So, the only feasible option left is to use heuristic algorithms. In this article, we apply and investigate a new heuristic approach called Heart algorithm to solve Travelling Salesman Problem. It simulates the heart action and circulatory system procedure in the human beings for searching the problem space. The Heart algorithm has the advantages of strong robustness, fast convergence, fewer setting parameters and simplicity. The results of the Heart algorithm on several standard TSP instances is compared with the results of the PSO and ACO algorithms and show that the Heart algorithm performs well in finding the shortest distance within the minimum span of time.

Hybrid discrete artificial bee colony algorithm with threshold acceptance criterion for traveling salesman problem

Artificial bee colony (ABC) algorithm, which has explicit strategies to balance intensification and diversification, is a smart swarm intelligence algorithm and was first proposed for continuous optimization problems. In this paper, a hybrid discrete ABC algorithm, which uses acceptance criterion of threshold accepting method, is proposed for Traveling Salesman Problem (TSP). A new solution updating equation, which can learn both from other bees and from features of problem at hand, is designed for the TSP. Aiming to enhance its ability to escape from premature convergence, employed bees and onlooker bees use threshold acceptance criterion to decide whether or not to accept newly produced solutions. Systematic experiments were performed to show the advantage of the new solution updating equation, to verify the necessity of using non-greedy acceptance strategy for keeping sufficient diversity, to compare different selection schemes for onlooker bees, and to analyze the contribution of scout bee. Comparison experiments performed on a wide range of benchmark TSP instances have shown that the proposed algorithm is better than other ABC-based algorithms and is better than or competitive with many other state-of-the-art algorithms.

A Hybrid Particle Swarm Optimization Method for Traveling Salesman Problem

Traveling salesman problem (TSP) is one well-known NP-Complete problem. The objective is to search the optimal Hamiltonian circuit (OHC) in a tourist map. The particle swarm optimization (PSO) integrated with the four vertices and three lines inequality is introduced to detect the OHC or approximate OHC. The four vertices and three lines inequality is taken as local heuristics to find the local optimal paths composed of four vertices and three lines. Each of this kind of paths in the OHC or approximate OHC conforms to the inequality. The particle swarm optimization is used to search an initial approximation. The four vertices and three lines inequality is applied to convert all the paths in the approximation into the optimal paths. Then a better approximation is obtained. The method is tested with several Euclidean TSP instances. The results show that the much better approximations are searched with the hybrid PSO. The convergence rate is also faster than the traditional PSO under the same preconditions.

Discrete symbiotic organisms search algorithm for travelling salesman problem

Abstract A Discrete Symbiotic Organisms Search (DSOS) algorithm for finding a near optimal solution for the Travelling Salesman Problem (TSP) is proposed. The SOS is a metaheuristic search optimization algorithm, inspired by the symbiotic interaction strategies often adopted by organisms in the ecosystem for survival and propagation. This new optimization algorithm has been proven to be very effective and robust in solving numerical optimization and engineering design problems. In this paper, the SOS is improved and extended by using three mutation-based local search operators to reconstruct its population, improve its exploration and exploitation capability, and accelerate the convergence speed. To prove that the proposed solution approach of the DSOS is a promising technique for solving combinatorial problems like the TSPs, a set of benchmarks of symmetric TSP instances selected from the TSPLIB library are used to evaluate its performance against other heuristic algorithms. Numerical results obtained show that the proposed optimization method can achieve results close to the theoretical best known solutions within a reasonable time frame.

A massively parallel neural network approach to large-scale Euclidean traveling salesman problems

This paper proposes a parallel computation model for the self-organizing map (SOM) neural network applied to Euclidean traveling salesman problems (TSP). This model is intended for implementation on the graphics processing unit (GPU) platform. The Euclidean plane is partitioned into an appropriate number of cellular units, called cells, and each cell is responsible of a certain part of the data and network. Compared to existing GPU implementations of optimization metaheuristics, which are often based on data duplication or mixed sequential/parallel solving, the advantage of the proposed model is that it is decentralized and based on data decomposition. Designed for handling large-scale problems in a massively parallel way, the required computing resources grow linearly along the problem size. Experiments are conducted on 52 publicly available Euclidean TSP instances with up to 85,900 cities for the largest TSPLIB instance and 71,009 cities for the largest National TSP instance. Experimental results show that our GPU implementations of the proposed model run significantly faster than the currently best-performing neural network approaches, to obtain results of similar quality.

Efficient preprocessing methods for tabu search: an application on asymmetric travelling salesman problem

This paper presents efficient methods of combining preprocessing methods and tabu search metaheuristic for solving large instances of the asymmetric travelling salesman problem (ATSP) with a focus on applications which require one to solve repeatedly different instances of ATSP and where for each instance one needs a reasonably good-quality solution quickly. For such applications, we present two hybrid metaheuristics, namely GA-SAG and RGC-SAG that, respectively, use genetic algorithm (GA) and randomized greedy contract (RGC) algorithm as preprocessing mechanisms, to sparsify a dense graph and apply an implementation of tabu search specifically designed for sparse asymmetric graphs (SAG) to further improve the solution quality. Our computational experience shows that both GA-SAG and RGC-SAG clearly outperform the conventional implementation of pure tabu search. Moreover, for benchmark instances, RGC-SAG reaches a solution within 1%–5% of the optimal solution much faster than the best known heuristics on benchmark problem instances. RGC-SAG provides tour values better than those obtained by PATCH or KP heuristic on 50% and 75% of the benchmark instances, respectively. Although the quality of the solutions obtained in Helsgaun or in the paper by doubly rooted stem and cycle ejection chain algorithm is marginally better than RGC-SAG on most of the benchmark instances, RGC-SAG establishes its potential with a significant reduction in computational time.

Simulated annealing based symbiotic organisms search optimization algorithm for traveling salesman problem

Symbiotic Organisms Search (SOS) algorithm is an effective new metaheuristic search algorithm, which has recently recorded wider application in solving complex optimization problems. SOS mimics the symbiotic relationship strategies adopted by organisms in the ecosystem for survival. This paper, presents a study on the application of SOS with Simulated Annealing (SA) to solve the well-known traveling salesman problems (TSPs). The TSP is known to be NP-hard, which consist of a set of (n−1)!/2 feasible solutions. The intent of the proposed hybrid method is to evaluate the convergence behaviour and scalability of the symbiotic organism's search with simulated annealing to solve both small and large-scale travelling salesman problems. The implementation of the SA based SOS (SOS-SA) algorithm was done in the MATLAB environment. To inspect the performance of the proposed hybrid optimization method, experiments on the solution convergence, average execution time, and percentage deviations of both the best and average solutions to the best known solution were conducted. Similarly, in order to obtain unbiased and comprehensive comparisons, descriptive statistics such as mean, standard deviation, minimum, maximum and range were used to describe each of the algorithms, in the analysis section. The Friedman's Test (with post hoc tests) was further used to compare the significant difference in performance between SOS-SA and the other selected state-of-the-art algorithms. The performances of SOS-SA and SOS are evaluated on different sets of TSP benchmarks obtained from TSPLIB (a library containing samples of TSP instances). The empirical analysis’ results show that the quality of the final results as well as the convergence rate of the new algorithm in some cases produced even more superior solutions than the best known TSP benchmarked results.

Minimization and maximization versions of the quadratic travelling salesman problem

The travelling salesman problem (TSP) asks for a shortest tour through all vertices of a graph with respect to the weights of the edges. The symmetric quadratic travelling salesman problem (SQTSP) associates a weight with every three vertices traversed in succession. If these weights correspond to the turning angles of the tour, we speak of the angular-metric travelling salesman problem (Angle TSP). In this paper, we first consider the SQTSP from a computational point of view. In particular, we apply a rather basic algorithmic idea and perform the separation of the classical subtour elimination constraints on integral solutions only. Surprisingly, it turns out that this approach is faster than the standard fractional separation procedure known from the literature. We also test the combination with strengthened subtour elimination constraints for both variants, but these turn out to slow down the computation. Secondly, we provide a completely different, mathematically interesting MILP linearization for the Angle TSP that needs only a linear number of additional variables while the standard linearization requires a cubic one. For medium-sized instances of a variant of the Angle TSP, this formulation yields reduced running times. However, for larger instances or pure Angle TSP instances, the new formulation takes more time to solve than the known standard model. Finally, we introduce MaxSQTSP, the maximization version of the quadratic travelling salesman problem. Here, it turns out that using some of the stronger subtour elimination constraints helps. For the special case of the MaxAngle TSP, we can observe an interesting geometric property if the number of vertices is odd. We show that the sum of inner turning angles in an optimal solution always equals . This implies that the problem can be solved by the standard ILP model without producing any integral subtours. Moreover, we give a simple constructive polynomial time algorithm to find such an optimal solution. If the number of vertices is even, the optimal value lies between 0 and and these two bounds are tight, which can be shown by an analytic solution for a regular n-gon.

A fireworks algorithm for solving travelling salesman problem

In this paper, a novel swarm intelligence algorithm inspired by observing fireworks explosions, called fireworks algorithm (FW), is proposed for solving the travelling salesman problem (TSP). The TSP is a well-known NP-hard combinatorial optimisation problem. The problem is easy to state, but hard to solve. Many real-world problems can be formulated as instances of the TSP, for example, computer wiring, vehicle routing, crystallography, robot control, drilling of printed circuit boards and chronological sequencing. The proposed algorithm has been performed on TSP instances taken from TSPLIB library and has been compared with other methods in the literature. Computational results showed that the proposed firework algorithm is competitive in terms of quality of the solutions compared to other techniques.

An Improved-time Polynomial-space Exact Algorithm for TSP in Degree-5 Graphs

The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard optimization problems. Following a recent trend of research which focuses on developing algorithms for special types of TSP instances, namely graphs of limited degree, in an attempt to reduce a part of the time and space complexity, we present a polynomial-space branching algorithm for the TSP in an n-vertex graph with degree at most 5, and show that it has a running time of O∗(2.3500n), which improves the previous best known time bound of O∗(2.4723n) given by the authors (the 12th International Symposium on Operations Research and Its Application (ISORA 2015), pp.45–58, 2015). While the base of the exponent in the running time bound of our algorithm is greater than 2, it still outperforms Gurevich and Shelah’s O∗(4nnlog n) polynomial-space exact algorithm for the TSP in general graphs (SIAM Journal of Computation, Vol.16, No.3, pp.486–502, 1987). In the analysis of the running time, we use the measure-and-conquer method, and we develop a set of branching rules which foster the analysis of the running time.

Socio-cognitively inspired ant colony optimization

Recently we proposed an application of ant colony optimization (ACO) to simulate socio-cognitive features of a population, incorporating perspective-taking ability to generate differently acting ant colonies. Although our main goal was simulation, we took advantage of the fact that the quality of the constructed system was evaluated based on selected traveling salesman problem instances, and the resulting computing system became a metaheuristic, which turned out to be a promising method for solving discrete problems. In this paper, we extend the initial sets of populations driven by different perspective-taking inspirations, seeking both optimal configuration for solving a number of TSP benchmarks, at the same time constituting a tool for analyzing socio-cognitive features of the individuals involved. The proposed algorithms are compared against classic ACO, and are found to prevail in most of the benchmark functions tested.

Research on an Improved ACO Algorithm Based on Multi-Strategy for Solving TSP

Ant colony optimization (ACO) algorithm is a metaheuristic inspired by the behavior of real ants in their search for the shortest path to food sources. The ACO algorithm takes on these characteristics of robust, positive feedback distributed computing, easy fusing with other algorithms. But the basic ACO algorithm has some deficiencies of premature and stagnation phenomenon in the evolution process, and is easily trapped into local optimal solution. And it is difficult to explore other solutions in the neighbor space. So a improved ACO(DPSEMACO) algorithm based on dual population strategy, bi-directional dynamic adjust evaporation factor strategy of the pheromone and parallel strategy is proposed to solve the traveling salesman problem(TSP). In the DPSEMACO algorithm, the ants are divided into the two subpopulations by borrowing the mutual cooperation mechanism of biological community, which evolve separately and exchange information timely. The bi-directional dynamic adjusting evaporation factor strategy of the pheromone is used to change the corresponding path pheromone of different subpopulations in order to avoid to trap into a local optimum. The parallel strategy can avoid falling into a local optimum. And the DPSEMACO algorithm can expand the search space and improve the overall searching performance by repeated changing the pheromone of the each subpopulation and adaptive adjusting evaporation factor. Finally, in order to prove the optimization performance of the proposed DPSEMACO algorithm, some classic TSP instances are selected from the TSPLIB in this paper. And some existing methods are selected to compare the optimization performance with the proposed DPSEMACO algorithm. The experimental results demonstrate that the proposed DPSEMACO algorithm is feasible and effective in solving TSP, and takes on a good global searching ability and high convergence speed.

Binary Cockroach Swarm Optimization for Combinatorial Optimization Problem

The Cockroach Swarm Optimization (CSO) algorithm is inspired by cockroach social behavior. It is a simple and efficient meta-heuristic algorithm and has been applied to solve global optimization problems successfully. The original CSO algorithm and its variants operate mainly in continuous search space and cannot solve binary-coded optimization problems directly. Many optimization problems have their decision variables in binary. Binary Cockroach Swarm Optimization (BCSO) is proposed in this paper to tackle such problems and was evaluated on the popular Traveling Salesman Problem (TSP), which is considered to be an NP-hard Combinatorial Optimization Problem (COP). A transfer function was employed to map a continuous search space CSO to binary search space. The performance of the proposed algorithm was tested firstly on benchmark functions through simulation studies and compared with the performance of existing binary particle swarm optimization and continuous space versions of CSO. The proposed BCSO was adapted to TSP and applied to a set of benchmark instances of symmetric TSP from the TSP library. The results of the proposed Binary Cockroach Swarm Optimization (BCSO) algorithm on TSP were compared to other meta-heuristic algorithms.

Meta-learning to select the best meta-heuristic for the Traveling Salesman Problem: A comparison of meta-features

The Traveling Salesman Problem (TSP) is one of the most studied optimization problems. Various meta-heuristics (MHs) have been proposed and investigated on many instances of this problem. It is widely accepted that the best MH varies for different instances. Ideally, one should be able to recommend the best MHs for a new TSP instance without having to execute them. However, this is a very difficult task. We address this task by using a meta-learning approach based on label ranking algorithms. These algorithms build a mapping that relates the characteristics of those instances (i.e., the meta-features) with the relative performance (i.e., the ranking) of MHs, based on (meta-)data extracted from TSP instances that have been already solved by those MHs. The success of this approach depends on the quality of the meta-features that describe the instances. In this work, we investigate four different sets of meta-features based on different measurements of the properties of TSP instances: edge and vertex measures, complex network measures, properties from the MHs, and subsampling landmarkers properties. The models are investigated in four different TSP scenarios presenting symmetry and connection strength variations. The experimental results indicate that meta-learning models can accurately predict rankings of MHs for different TSP scenarios. Good solutions for the investigated TSP instances can be obtained from the prediction of rankings of MHs, regardless of the learning algorithm used at the meta-level. The experimental results also show that the definition of the set of meta-features has an important impact on the quality of the solutions obtained.