Solving the travelling salesman problem using Bloch sphere encoding

The Traveling Salesman Problem (TSP) is a combinatorial optimization problem which has a plethora of real-world applications in various domains. Since large scale TSP problems are difficult to be solved by deterministic algorithms in a reasonable amount of time, heuristic and meta-heuristic algorithms such as swarm intelligence algorithms are usually used for solving TSP. These algorithms provide near-optimal solutions in a feasible amount of time. The Dragonfly Algorithm (DA) is a recent swarm intelligence algorithm and it has shown to have a higher performance as compared to other swarm intelligence algorithms in various applications. Since the original DA algorithm is proposed for solving continuous optimization problems, it cannot be used for solving TSP and although a binary version of the algorithm, called BDA, is also proposed, it is not suitable for solving TSP. Hence, in this paper, a variant of the DA algorithm is proposed. DA is adapted for solving TSP by adapting its equations and making use of the method of swap sequences to update the position of the artificial dragonflies in the search space. The algorithm is employed to solve a TSP problem which consists of four cities and the results show that the proposed algorithm is able to provide the optimal TSP path.

The Traveling Salesman Problem (TSP) is an NP-hard graph search problem. Despite having numerous modifications of the original abstract problem, Time Dependent Traveling Salesman Problem (TD TSP) was one of the most realistic extensions under real traffic conditions. In TD TSP the edges between nodes are assigned higher costs (weights), if they were traveled during the rush hour periods, or crossed the traffic jam regions, such as the city center(s). In this paper we introduce an even more real-life motivated approach, the Intuitionistic Fuzzy Time Dependent Traveling Salesman Problem (IFTD TSP), which is a further extension of the TSP, and also of the classic TD TSP, with the additional notion of using intuitionistic fuzzy sets for the definition of uncertain costs, time, and space of the rush hour—traffic jam region affecting graph sections. In IFTD TSP we use fuzzy memberships and non-memberships sets for estimating the vague costs between nodes in order to quantify the behavior of traffic jam regions, and the rush hour periods. Since intuitionistic fuzzy sets are generalizations of classic fuzzy sets, our approach may be considered an extension and substitution of the original abstract TD TSP problem, even, of the (classic) Fuzzy TD TSP. Lastly, DBMEA (Discrete Bacterial Memetic Evolutionary Algorithm) was applied on the IFTD TSP model, the results of the simulation runs based on some extensions of the benchmarks generated from the original TD TSP data set showed quite good and promising preliminary results.

Results on human performance on the Traveling Salesman Problem (TSP) from different laboratories show high consistency. However, one exception is in the area of individual differences. While one research group has consistently failed to find systematic individual differences across instances of TSPs (Chronicle, MacGregor and Ormerod), another group (Vickers, Lee and associates) has found individual differences both within TSP performance and between TSP performance and other cognitive tasks. Among possible reasons for the conflicting results are differences in procedure and differences in the problem instances used. To try to resolve the discrepancy, we collected data on TSP performance by combining the procedure used by one group with problem instances used by the other. The comparison involved nine 30-node and nine 40-node TSP problems previously used by the Vickers group, using computer presentation. Here, we had the same problems completed by 112 participants using a paper-and-pencil mode of presentation. We examined the results in the form of distributions of correlations across individuals for each pair of problems of the same size. The distributions for the computer and paper forms of presentation were very similar, and centered between correlations of 0.20 and 0.30. The results indicated the presence of individual differences at a level that fell between those previously reported by the two laboratories. The pattern of results indicated that previous discrepancies did not arise because of differences in procedure. Instead, individual differences appeared to become more prevalent as the difficulty of problems increased. The results are consistent with an explanation that performance on simpler instances is dominated by lower-level processes, but that as instance difficulty increases, higher-level functions become increasingly involved.

In this article, we have introduced an advanced new method of solving a traveling salesman problem (TSP) with the whale optimization algorithm (WOA), and K-means which is a partitioning-based algorithm used in clustering. The whale optimization algorithm first was introduced in 2016 and later used to solve a TSP problem. In the TSP problem, finding the best path, which is the path with the lowest value in the fitness function, has always been difficult and time-consuming. In our algorithm, we want to find the best tour by combining it with K-means which is a clustering method. In other words, we want to divide our problem into smaller parts called clusters, and then we join the clusters based on their distances. To do this, the WOA algorithm, TSP, and K-means must be combined. Separately, the WOA-TSP algorithm which is an unclustered algorithm is also implemented to be compared with the proposed algorithm. The results are shown through some figures and tables, which prove the effectiveness of this new method.

As the amount and scale of cities in Traveling Salesman Problem (TSP) rise, the algorithmic complexity is exponentially increasing. The difficulty is how to design a suitable algorithm to solve large-scale TSPs. A Cooperative Co-evolutionary Ant Colony Optimization algorithm (CC-ACO) is proposed in this paper based on the concept of divide and conquer. The Iterative Self-Organizing Data Analysis (ISODATA) clustering algorithm tackles the problem by dividing it into a set of smaller and simpler sub-components and the ACO algorithms are designed for optimizing them separately. Numerical tests are then conducted to investigate algorithms, analyze results, and compare performances. The simulation findings show a significant efficiency of the suggested algorithm on the TSPLIB data set. Finally, we extend the large-scale TSP problem to the field of art and use the presented algorithm to optimize the path of discrete pixels in the picture, showing the artistic painting of the TSP art project.

Most discrete optimization problems belong to the NP-complete or NP-hard classes. A benchmark problem to assess the performance of new heuristics to solve these classes of problems is the traveling salesman problem (TSP). This paper proposes some modifications on a self-organizing network, named RABNET- TSP, to solve TSP problems. The modified algorithm is compared, with encouraging results, to other proposals from the literature on a number of different TSP instances.

This paper discuss about the role of Decision Support Tool in Travelling Salesman Problem (TSP) for helping the researchers who doing research in same area will get the better result from the proposed algorithm. A study has been conducted and Rapid Application Development (RAD) model has been use as a methodology which includes requirement planning, user design, construction and cutover. Water Flow Algorithm (WFA) with initialization technique improvement is used as the proposed algorithm in this study for evaluating effectiveness against TSP cases. For DST evaluation will go through usability testing conducted on system use, quality of information, quality of interface and overall satisfaction. Evaluation is needed for determine whether this tool can assists user in making a decision to solve TSP problems with the proposed algorithm or not. Some statistical result shown the ability of this tool in term of helping researchers to conduct the experiments on the WFA with improvements TSP initialization.

The increase in the use of parallel distributed architectures in order to solve large-scale scientific problems has generated the need for performance prediction for both deterministic applications and non-deterministic applications. The development of a new prediction methodology to estimate the execution time of a hard data-dependent parallel application that solves the traveling salesman problem (TSP) is the primary target of this study. It consists of two big stages: the execution of the TSP algorithms with different input data in order to collect useful data and the application of a data mining procedure through a KDD process. The approach makes it also possible to evaluate other practical problems that can be formulated as TSP problems. The experimental results are quite promising, the capacity of prediction is greater than 75%.

Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO) algorithms have attracted the interest of researchers due to their simplicity, effectiveness and efficiency in solving real world optimization problems. Swarm-inspired optimization has recently become very popular. Both ACO and PSO are successfully applied in the Traveling Salesman Problem (TSP). Our approach consists in combining Fuzzy Logic with ACO (FACO - Fuzzy Ant Colony Optimization) and PSO (FPSO - Fuzzy Particle Swarm Optimization) for solving the TSP. Experimental results and comparative studies illustrate the importance of Fuzzy logic in reducing the time and the best length for the TSP problems considered.

This paper is concerned with the discrete combinatorial optimization problem. A novel set-based differential evolution (SBDE) algorithm for traveling salesman problem (TSP) is proposed and evaluated. SBDE is based on a set-based representation scheme that enables SBDE to characterize the discrete search space of TSP. The candidate solution, vector and the arithmetical operator in SBDE are redefined. Numerical experiments are carried out using the TSPLIB of test TSP problems. The obtained results show that the proposed approach is effective in quality for solving TSP.

Genetic algorithms (GAs) are search algorithms based on population genetics and natural selection concepts. Maintaining population variety in GAs is critical for ensuring global exploration and mitigating the risks of premature convergence. Rapid convergence to local optima is one such challenge in the application of genetic algorithms. To address this issue, we provide Cave-Surface GA (CSGA), an alternative method based on the Dual Population Genetic Algorithm and inspired by the genetic variety observed in Mexican cavefish. Through inter-population crossbreeding, CSGA increases diversity via a secondary population (Cave population) and facilitates the exchange of information between populations, effectively counteracting premature convergence. Several experiments are carried out utilizing benchmark instances of the Traveling Salesman Problem (TSP) obtained from TSPLIB, a well-known TSP problem library. Our experimental results over many TSP instances show that CSGA outperforms both classic GAs and other GAs that use diversity preservation techniques, such as Multipopulation GA (MPGA). CSGA has the potential to give promising solutions to challenging optimization issues like TSP.

A parallel approach for solving a large-scale Traveling Salesman Problem (TSP) is presented. The problem is solved in four stages by using the following sequence of procedures: decomposing the input set of points into two or more clusters, solving the TSP for each of these clusters to generate partial solutions, merging the partial solutions to create a complete initial solution M0, and finally optimizing this solution. Lin-KernighanHelsgaun (LKH) algorithm is used to generate the partial solutions. The main goal of this research is to achieve speedup and good quality solutions by using parallel calculations. A clustering algorithm produces a set of small TSP problems that can be executed in parallel to generate partial solutions. Such solutions are merged to form a solution, M0, by applying the ―Ring‖ method. A few optimization algorithms were proposed to improve the quality of M0 to generate a final solution Mf. The loss of quality of the solution by using the developed approach is negligible when compared to the existing best-known solutions but there is a significant improvement in the runtime with the developed approach. The minimum number of processors that are required to achieve the maximum speedup is equal to the number of clusters that are created.

The traveling-salesman problem (TSP) for a class of carrier-vehicle systems in which a slow carrier with infinite operating range cooperates with a faster vehicle that has a limited operating range, was studied. A heuristic solution based on the Euclidean TSP (E-TSP) has been proposed and an analytical bound on its conservativeness has been derived. One of the main feature of this class of TSP problems is that, although still NP-hard, they admit polynomial-in-time optimization schemes. The CV-TSP heuristic proposed consists of two steps. First, it determines the visiting order of the almost-optimal E-TSP tour for the set of given points, and second, it uses the visiting order to solve the resulting visit of n points via the convex optimization formulation. The targets to be visited are limited to five randomly generated points that still make the computation of the exact optimal solution in a reasonable time possible.

Ant Colony Optimization (ACO) is a metaheuristic inspired by the foraging behavior of ant colonies that has been successful in the resolution of hard combinatorial optimization problems like the Traveling Salesman Problem (TSP). This paper proposes the Omicron ACO (OA), a novel population-based ACO alternative originally designed as an analytical tool. To experimentally prove OA advantages, this work compares the behavior between the OA and the MMAS as a function of time in two well-known TSP problems. A simple study of the behavior of OA as a function of its parameters shows its robustness.

Ant Colony Optimization (ACO) is a well-established metaheuristic successfully applied to solve hard combinatorial optimization problems, including Travelling Salesman Problem (TSP). However, ACO algorithm as many population-based approaches has some disadvantages, such as lower solution quality and longer computational time. To overcome these issues, parallel Cultural Ant Colony Optimization (pCACO) is introduced in this paper. The proposed approach hybridises Cultural Algorithm with ACO-based \(\mathcal {MAX}\)-\(\mathcal {MIN}\) Ant System. pCACO has been implemented on Graphics Processing Units (GPUs) using CUDA programming model. Through testing nine benchmark asymmetric TSP problems, the experimental results show the new method enhances the solution quality when compared to sequential and parallel ACO, yielding comparable computational time to parallel ACO.