A Comparative Study of Three Nature-Inspired Algorithms Using the Euclidean Travelling Salesman Problem

Travelling Salesman Problem(TSP) is an NP hard problem, which has been addressed via several heuristic algorithms like genetic algorithms (GA) etc. The paper presents TSP solution using Teaching Learning Based optimization(TLBO) which includes many similarities like the other nature-inspired algorithms. In addition, the strategy adds an auxiliary operator to regulate the sequence of array elements in mutation process with partial-mapped crossover used over the selected candidates of the population. Experimental study shows that introduced strategy can provide better convergence. The prominent feature of the strategy is in its stability and superior ways to handle computational complexity while dealing with TSP optimization as compared to GA.

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.

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

<p><span>Travelling salesman problem is a well researched problem in computer science and has many practical applications. It is classified as a NP-hard problem as its exact solution can only be obtained in exponential time unless P = NP. There are different variants of the travelling salesman problem (TSP) and in this paper, asymmetric travelling salesman problem is addressed since this variant is quite often observed in real world scenarios. There are a number of heuristic approaches to this problem which provides approximate solutions in polynomial time, however this paper proposes an exact optimal solution which is accelerated with the help of multi-threading-based parallelization. In order to find the exact optimal solution, we have used the held-karp algorithm involving dynamic programming and to reduce the time taken to find the optimal path, we have used a multi-threaded approach to parallelize the processing of sub-problems by leveraging the central processing unit cores (CPUs). This method is an extension of a well researched solution to the TSP; however, this method shows that solutions to computationally intensive problems involving sub-problems such as the asymmetic travelling salesman problem (ATSP) can be accelerated with the help of modern CPUs.</span></p>

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.

Discrete Spider Monkey Optimization for Travelling Salesman Problem

Metaheuristic algorithms, such as simulated annealing, tabu search and evolutionary algorithms, are popular techniques for solving optimization problems when exact methods are not practical. For example, the run times required to obtain exact solutions to many common combinatorial problems grow exponentially (or worse) with the problem size, and as a result, solving even a relatively modest sized problem of this type may require many centuries of computation time, even on the fastest computer of the day. Such problems are known collectively as NP-Hard, and include the travelling salesman problem (TSP), which is probably the best known problem in the class. The present study will concentrate on a type of NP-Hard combinatorial problem known as the set partitioning problem. If we have n objects to partition into m sets, in such a way that each object must be assigned to exactly one set, it follows that there are m n different ways that the n objects can be assigned to the m sets, for a straightforward unconstrained problem. It is instructive to note that every time the problem size of the set partitioning problem is increased by one object, the corresponding run time for an exhaustive search algorithm will increase by a factor of m, and thus the run time grows exponentially as the number of objects – n – increases. While it is true that much better exact methods than exhaustive search have been developed for most NP-Hard problems, the ‘growth factor’ remains exponential for the run time, and no one in history has so far managed to change that.

Generalized traveling salesman problem (GTSP) is an extension of classical traveling salesman problem (TSP), which is a combinatorial optimization problem and an NP-hard problem. In this paper, an efficient discrete state transition algorithm (DSTA) for GTSP is proposed, where a new local search operator named \textit{K-circle}, directed by neighborhood information in space, has been introduced to DSTA to shrink search space and strengthen search ability. A novel robust update mechanism, restore in probability and risk in probability (Double R-Probability), is used in our work to escape from local minima. The proposed algorithm is tested on a set of GTSP instances. Compared with other heuristics, experimental results have demonstrated the effectiveness and strong adaptability of DSTA and also show that DSTA has better search ability than its competitors.

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.

Vehicle routing problems (VRPs) are a fundamental class of combinatorial optimization problems (COPs) in computer science and operations research, with diverse real-world applications in logistics, transportation, and manufacturing. The intrinsic NP-hard nature makes VRPs exponentially expensive to be solved by exact solvers. As an alternative, heuristic solvers deliver suboptimal solutions within reasonable time but require extensive hand-crafted rules and domain expertise tailored to each specific problem. Recently, neural combinatorial optimization (NCO), which leverages machine learning to learn heuristics in a data-driven manner, has gained significant attention. These neural solvers demonstrate strong performance while reducing computational overhead and reliance on domain expertise compared to traditional solvers. However, they face significant generalization challenges. For example, their performance may degrade substantially when applied to instances with different data distributions, problem scales, or constraints than those encountered during training. This generalization issue severely limits their practical applicability. This thesis aims to systematically address these challenges and advance the development of generalizable neural solvers for VRPs. The thesis begins by enhancing the cross-distribution generalization of neural VRP solvers through an adversarial training approach. We introduce an ensemble-based Collaborative Neural Framework (CNF) that adversarially trains multiple models in a collaborative manner, promoting robustness against adversarial attacks while also boosting performance on clean instances. In this framework, an attacker generates hard instances by perturbing the data distribution, challenging the models to adapt and thereby strengthening cross-distribution generalization. Additionally, a neural router is designed to efficiently distribute training instances among the models, thereby improving load balancing and collaborative efficacy. Extensive experiments on two classic routing problems -- the travelling salesman problem (TSP) and the capacitated vehicle routing problem (CVRP) -- demonstrate the effectiveness and versatility of CNF in enhancing the cross-distribution generalization. Next, the thesis further expands neural solvers to simultaneously address both cross-size and cross-distribution generalization challenges. We propose Omni-VRP, a generic meta-learning framework that enables effective training of an initial model with the capacity for rapid adaptation to new tasks during inference, where each task corresponds to a set of instances with the same problem scale and data distribution. Additionally, we introduce a simple yet efficient approximation method to reduce the training overhead associated with second-order derivative calculations. Extensive experiments on both synthetic and benchmark instances of TSP and CVRP validate the effectiveness of our approach. Finally, the thesis pioneers the study of cross-problem generalization, pushing the boundaries of neural VRP solvers in learning generalizable representations across diverse constraints. We propose MVMoE, a unified neural solver capable of solving 16 VRP variants simultaneously in a zero-shot manner through attribute composition. To efficiently enhance model capacity, we employ a mixture-of-experts architecture with a hierarchical gating mechanism, striking a balance between empirical performance and computational complexity. Experimentally, our method significantly promotes zero-shot generalization on 10 unseen VRP variants, and showcases decent results on the few-shot setting and real-world benchmark instances. Overall, this thesis represents a pioneering exploration and substantial advancement in developing generalizable neural solvers for VRPs. It investigates innovative designs across various components, including model architectures, training algorithms, and problem settings, to enhance the generalization of neural solvers across diverse contexts such as varying data distributions, problem scales, and constraints. These contributions encourage neural solvers to learn more robust and generalizable representations, paving the way for the first generation of foundation models capable of addressing a wide range of COPs. Ultimately, the insights gained here not only propel the field of NCO forward but also enrich the broader landscape of learning-based optimization methods.

Optimization problem like Travelling Salesman Problem (TSP) can be solved by applying Genetic Algorithm (GA) to obtain perfect approximation in time. In addition, TSP is considered as a NP-hard problem as well as an optimal minimization problem. Selection, crossover and mutation are the three main operators of GA. The algorithm is usually employed to find the optimal minimum total distance to visit all the nodes in a TSP. Therefore, the research presents a new crossover operator for TSP, allowing the further minimization of the total distance. The proposed crossover operator consists of two crossover point selection and new offspring creation by performing cost comparison. The computational results as well as the comparison with available well-developed crossover operators are also presented. It has been found that the new crossover operator produces better results than that of other cross-over operators.

The article considers the solution of the traveling salesman problem using reinforcement learning. This task is to find the optimal route that passes through the specified cities at least once and then returns to the original city. The traveling salesman problem belongs to the class of NP-hard problems, and therefore finding an exact solution for sufficiently large data sets using classical mathematical methods turns out to be impossible in a reasonable time. The aim of the work is to increase the efficiency and speed of solving the traveling salesman problem with a large number of vertices by using the reinforcement learning method. A variant of solving the traveling salesman problem using the Q-learning algorithm is considered, which allows the agent to learn the optimal behavior strategy in the process of interacting with the environment, which is a set of cities and roads between them, which are modeled as a graph. Within the framework of this study, a software package was developed to solve the traveling salesman problem using reinforcement learning. A series of experiments related to the appearance of unforeseen conditions complicating the work of a traveling salesman, such as the appearance of "problem" zones or zones, the passage through which gives a reduction in time. The results of the experiments showed the capabilities of the developed algorithm for adaptation to changes and obstacles of the environment. The resulting algorithm shows itself to be more stable and reliable compared to the algorithms obtained by classical solution methods. The solution of this problem can be used in practice in a variety of real situations, including optimization of deliveries or network routing, assistance in the work of logistics departments of various transport, postal and courier services.

......... Non-deterministic polynomial time (commonly termed ‘NP-complete’) problems are relevant to many computational tasks of practical interest—such as the ‘travelling salesman problem’—but are difficult to solve: the computing time grows exponentially with problem size in the worst case. It has recently been shown that these problems exhibit ‘phase boundaries’, across which dramatic changes occur in the computational difficulty and solution character—the problems become easier to solve away from the boundary. Here we report an analytic solution and experimental investigation of the phase transition in K-satisfiability, an archetypal NP-complete problem. Depending on the input parameters, the computing time may grow exponentially or polynomially with problem size; in the former case, we observe a discontinuous transition, whereas in the latter case a continuous (second-order) transition is found. The nature of these transitions may explain the differing computational costs, and suggests directions for improving the efficiency of search algorithms. Similar types of transition should occur in other combinatorial problems and in glassy or granular materials, thereby strengthening the link between computational models and properties of physical systems. Many computational tasks of practical interest are surprisingly difficult to solve even using the fastest available machines. Such problems, found for example in planning, scheduling, machine learning, hardware design, and computational biology, generally belong to the class of NP-complete problems 1‐3 . NP stands for ‘nondeterministic polynomial time’, which denotes an abstract computational model with a rather technical definition. Intuitively speaking, this class of computational tasks consists of problems for which a potential solution can be checked efficiently for correctness, yet finding such a solution appears to require exponential time in the worst case. A good analogy can be drawn from mathematics: proving open conjectures in mathematics is extremely difficult, but verifying any given proof (or solution) is generally relatively straightforward. The class of NP-complete problems lies at the foundations of the theory of computational complexity in modern computer science. Literally thousands of computational problems have been shown to be NP-complete. The completeness property of NPcomplete problems means that if an efficient algorithm for solving just one of these problems could be found, one would immediately have an efficient algorithm for all NP-complete problems. However,

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

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.