Anytime Behavior of Inexact TSP Solvers and Perspectives for Automated Algorithm Selection

Traveling salesman problem (TSP) means that a travelling salesman needs to promote products in n cities (including the city where he lives). After visiting each city (each city can be visited once), he returns to the departure city. Let’s suppose that there is one road to connect each two cities. What is the best route to follow in order to minimize the distance of the journey? TSP has been proven to be a NP-hard problem, i.e. failure of finding a polynomial time algorithm to get a optimal solution. TSP is easy to interpret, yet hard to solve. This problem has aroused many scholars’ interests since it was put forward in 1932. However, until now, no effective solution has been found. Though TSP only represents a problem of the shortest ring road, in actual life, many physical problems are found to be the TSP. Example 1, postal route. Postal route problem is a TSP. Suppose that a mail car needs to collect mails in n places. Under such circumstances, you can show the route through a drawing containing n+1 crunodes. One crunode means a post office which this mail car departures from and returns to. The remaining n crunodes mean the crunodes at which the mails need to be collected. The route that the mail car passes through is a travelling route. We hope to find a travelling route with the shortest length. Example 2, mechanical arm. When a mechanical arm is used to fasten the nuts for the ready-to-assembling parts on the assembly line, this mechanical arm will move from the initial position (position where the first nut needs to be fastened) to each nut in proper order and then return to the initial position. The route which the mechanical arm follows is a travelling route in the drawing which contains crunodes as nuts; the most economical travelling route will enable the mechanical arm to finish its work within the shortest time. Example 3, integrated circuit. In the course of manufacturing the integrated circuit, we often need to insert thousands of electrical elements. It will consume certain energy when moving from one electrical element to the other during manufacturing. How can we do to arrange the manufacturing order to minimum the energy consumption? This is obviously a solution for TSP. Except for the above examples, problems like route distribution of transportation network, choice of tourist route, laying of pipelines needed for city planning and engineering construction are interlinked with the problems of finding the shortest route. So, it is of significance to make a study on the problem of the shortest route. This renders us a use value. As finding a solution for TSP plays an important role in the real life, since the TSP appeared, it has attracted many scholars to make a study on it.

The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard combinatorial optimization problems, widely studied due to its numerous industrial applications in logistics, supply chain management, and resource planning. Over the years, various metaheuristic algorithms have been developed to provide near-optimal solutions within a reasonable computational time. While traditional methods such as Genetic Algorithms (GA) and Simulated Annealing (SA) have been applied widely to the TSP, recent developments in metaheuristics have introduced new methods that can ensure improved performance. In this study, we compare three new metaheuristics: Arctic Puffin Optimization (APO), Human Evolutionary Optimization (HEO), and Ship Rescue Optimization (SRO). These methods, inspired by puffin's behavior, human evolutions, and naval rescue, respectively, have demonstrated high optimization capability in various areas. However, these have not been extensively tried on solving the TSP yet. This research seeks to establish the performance of these algorithms by comparing their convergence rate, solution quality, and computational efficiency with well-established metaheuristics. The results of the experiment expose the strengths and weaknesses of such innovative approaches, showing their potential as competitive alternatives to solve complex combinatorial optimization problems

The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard combinatorial optimization problems (COPs). Many social production problems can be effectively represented as instances of TSPs. However, solving large-scale TSPs remains a significant challenge for conventional Von Neumann computers. Many studies have proposed annealing processors to address large-scale COPs, but most of them focus on unconstrained problems, such as the Maxcut problem. In this paper, a scalable decoupled-clustering annealng processor (DCAP) for efficiently handling large-scale TSPs is presented. A decoupled hierarchical clustering algorithm is proposed for higher convergence speed and improved scalability. Several techniques have been developed in hardware to minimize area overhead and processing time, including a modified spin connection topology for the Ising model, an area-efficient random threshold generator, a one-step spin update scheme and a dynamic prediction method. The DCAP prototype is implemented on FPGA with an operating frequency of 125MHz. We tested our design on various TSP instances from the TSPLIB. Results show that our design outperforms the CPU- and GPU-based Neuro-Ising scheme by achieving maximum speedups of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$780\times $ </tex-math></inline-formula> and a 42% improvement in accuracy. With multi-chip interconnection, DCAP is able to handle problems of scale up to 85900 cities.

Travelling Salesman Problem with profits (TSPP) is a special case of the general Travelling Salesman Problem, all nodes must not be visited, but profit is collected from visited nodes. It is a well-known NP-hard combinatorial optimization problem in the literature. Because of the problem's complexity, exact methods cannot find the global optimum solution to this problem, so many heuristic and metaheuristic solution techniques are studied to find a feasible solution in a reasonable time. In this research, two different metaheuristic algorithms, Simulated Annealing with Many-moves and Variable Neighborhood Search, are proposed to solve the TSPP. Proposed algorithms are tested with three different problem instances and compared in terms of the efficiency of algorithms.

Traveling salesman problem (TSP) is a well-known NP-hard combinatorial optimization problem. It has been solved by a number of exact and approximate algorithms and serves as a testbed for new heuristic and metaheuristic optimization algorithms. However, it is often not easy to evaluate the hardness (complexity) of a TSP instance. Simple measures such as the number of cities or the minimum (maximum) route length do not capture the internal structure of a TSP instance sufficiently. In this work, we propose a new method for the assessment of TSP instance complexity based on clustering. The new approach is evaluated on a set of randomized TSP instances with different structure and its relation to the performance of a selected metaheuristic TSP solver is studied.

Discrete Spider Monkey Optimization for Travelling Salesman Problem

Recently, the nature has become a source of inspiration for the creation of many algorithms. A great research effort has been devoted to the development of new metaheuristics, especially nature-inspired one to solve numerous difficult combinatorial problems appearing in various industrial, economic, and scientific domains. The nature-inspired algorithms offer additional advantages over classical algorithms; they seek to find acceptable results within a reasonable time, rather than an ability to guarantee the optimal or sub-optimal solution. The travelling salesman problem (TSP) is an important issue in the class of combinatorial optimization problem and also classified as NP-hard problem and no polynomial time algorithm is known to solve it. Based on three nature-inspired algorithms, this paper proposes a comparative study to solve TSP. The proposed algorithms are evaluated on a set of symmetric benchmark instances from the TSPLIB library.

Various neural network models have been proposed to tackle combinatorial optimization problems such as the travelling salesman problem (TSP). Existing learning-based TSP methods adopt a simple setting that the training and testing data are independent and identically distributed. However, the existing literature fails to solve TSP instances when training and testing data have different distributions. Concretely, we find that different training and testing distribution will result in more difficult TSP instances, i.e., the solution obtained by the model has a large gap from the optimal solution. To tackle this problem, in this work, we study learning-based TSP methods when training and testing data have different distributions using adaptive-hardness, i.e., how difficult a TSP instance can be for a solver. This problem is challenging because it is non-trivial to (1) define hardness measurement quantitatively; (2) efficiently and continuously generate sufficiently hard TSP instances upon model training; (3) fully utilize instances with different levels of hardness to learn a more powerful TSP solver. To solve these challenges, we first propose a principled hardness measurement to quantify the hardness of TSP instances. Then, we propose a hardness-adaptive generator to generate instances with different hardness. We further propose a curriculum learner fully utilizing these instances to train the TSP solver. Experiments show that our hardness-adaptive generator can generate instances ten times harder than the existing methods, and our proposed method achieves significant improvement over state-of-the-art models in terms of the optimality gap. The codes are publicly available.

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.

Traveling Salesman Problem (TSP) is a well-known NP-hard combinatorial optimization problem. The Particle Swarm Optimization has been proven to succeed in lots of problems, but the PSO algorithm is challenging due to a variety of factors such as easy to fall into local optimal solution and the convergence speed is slow in the later. In this paper, we propose an adaptive mutation multi-particle swarm optimization algorithm (AMPSO) to the TSP. The experimental results show that the proposed algorithm can achieves better performance compared to the standard PSO method to solve the TSP.

The travelling salesman problem (TSP) with neighbourhoods extends the TSP to the case where each vertex of the tour is allowed to move in a given region. This NP-hard optimization problem has recently received increasing attention in several technical fields such as robotics, unmanned aerial vehicles, or utility management. In this paper, the problem is formulated as a non-convex mixed-integer nonlinear programme (MINLP) having the property that fixing all the integer variables to any integer values yield a convex nonlinear programme. This property is used to modify the global MINLP optimizer Couenne, improving by orders of magnitude its performance and allowing the exact solution of instances large enough to be useful in applications. Computational results are presented where neighbourhoods are either polyhedra or ellipsoids in ℝ2 or ℝ3 and with the Euclidean norm as distance metric.

In view of Traveling Salesman Problem (TSP) is a NP hard problem, Lion Swarm Optimization (LSO) Algorithm is a new swarm intelligence optimization algorithm. Therefore, in this paper, it uses improved hybrid LSO algorithm to solve TSP. Because the basic LSO algorithm is mainly used to solve the continuous problem, and the TSP is a discrete combinatorial optimization problem, this paper improves the LSO algorithm, introduces the greedy strategy, and combines with the Population Competitive Search (PCS) algorithm in the later stage of the search. As a result, the convergence speed is accelerated and the search efficiency is improved. The classic TSP example is tested, and the simulation results show that the algorithm proposed in this paper has strong stability and is an effective method for solving TSP.

The Traveling Salesman Problem (TSP) is one of the most widely discussed problems in combinatorial optimization. It has many practical applications in fields of distribution and logistics management, scheduling problems etc. Since these problems are hard, in addition to exact algorithms, research has focused on heuristic techniques to solve TSPs. Computational time is a major concern while solving large TSPs. This problem intensifies further if the graph becomes asymmetric (ATSP). Metaheuristics like tabu search are widely used to find a reasonably good tour fast. Given the practical relevance of ATSPs the lack of literature on it is surprising. The primary objective of our work is to implement tabu search for large ATSPs to obtain good tours in reasonable time. To do that, we make the underlying graph sparse by developing an elite tour based preprocessing scheme. Tabu search is implemented on this reduced graph which results in a reduction of computational time. We also create diversified initial tours suitable for multi-start tabu search in this process. We present our computational experiences both on randomly generated instances and benchmark instances.

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.

Travelling Salesman Problem (TSP) is a NP-Hard combinatorial optimization problem and many real life problems are constrained replica of it which possesses exponential time complexity and requires heavy combination capability. In this work a new nature inspired meta-heuristics called Egyptian Vulture Optimization Algorithm (EVOA) is being introduced and presented for the first time and illustrated with examples how it can be utilized for the constrained graph based problems and is utilized to solve the various dimensional datasets of the traditional travelling salesman problem. There are not many discrete optimization bio-inspired algorithms available in the literature and in that respect it is a novel one which can readily utilized for the graph based and assignment based problems. This EVOA is inspired by the natural and skilled phenomenal habits, unique perceptions and intelligence of the Egyptian Vulture bird for carry out the livelihood and acquisition of food which is inevitable for any kind of organisms. The Egyptian Vulture bird is one of the few birds who are known for showing dexterous capability when it comes to its confrontation with tough challenges and usage of tools with combinations of force and weakness finding ability. The results show that the Egyptian Vulture Optimization meta-heuristics has potential for deriving solutions for the TSP combinatorial problem and it is found that the quality and perfection of the solutions for the datasets depend mainly on the number of dimensions when considerable for the same number of iterations.Keywordsegyptian vulture optimization algorithmdiscrete domain problemscombinatorial optimizationgraph based constrained problemsnature inspired heuristicstravelling salesman problem