Traveling Salesman Problem Problem Research Articles

Visual Reasoning and Multi-Agent Approach in Multimodal Large Language Models (MLLMs): Solving TSP and mTSP Combinatorial Challenges

Multimodal Large Language Models (MLLMs) harness comprehensive knowledge spanning text, images, and audio to adeptly tackle complex problems. This study explores the ability of MLLMs in visually solving the Traveling Salesman Problem (TSP) and Multiple Traveling Salesman Problem (mTSP) using images that portray point distributions on a two-dimensional plane. We introduce a novel approach employing multiple specialized agents within the MLLM framework, each dedicated to optimizing solutions for these combinatorial challenges. We benchmarked our multi-agent model solutions against the Google OR tools, which served as the baseline for comparison. The results demonstrated that both multi-agent models—Multi-Agent 1, which includes the initializer, critic, and scorer agents, and Multi-Agent 2, which comprises only the initializer and critic agents—significantly improved the solution quality for TSP and mTSP problems. Multi-Agent 1 excelled in environments requiring detailed route refinement and evaluation, providing a robust framework for sophisticated optimizations. In contrast, Multi-Agent 2, focusing on iterative refinements by the initializer and critic, proved effective for rapid decision-making scenarios. These experiments yield promising outcomes, showcasing the robust visual reasoning capabilities of MLLMs in addressing diverse combinatorial problems. The findings underscore the potential of MLLMs as powerful tools in computational optimization, offering insights that could inspire further advancements in this promising field.

ОЦІНКА ОБЧИСЛЮВАЛЬНОЇ СКЛАДНОСТІ ГЕНЕТИЧНОГО АЛГОРИТМУ

The article is devoted to the estimation of computational complexity of a genetic algorithm as one of the key tools for solving optimisation problems. The theoretical aspects of computational complexity of algorithms and the interrelation of elements of a genetic algorithm are considered. The main types of computational complexity of algorithms are described: time, simple and asymptotic. Five basic rules for calculating the asymptotic complexity are given. A mathematical apparatus for estimating the asymptotic complexity of a genetic algorithm is presented, which takes into account the costs of forming the initial population and performing evolution. Evolution occurs through iterations, during which generations of individuals are subjected to certain operations in order to find an optimal solution (crossing, mutation, chromosome decoding, etc.). GA, as a global search algorithm, is considered to find the optimal path without getting stuck in local minima. To assess the computational complexity of GA, we consider solving the traveling salesman problem (TSP) for 28 cities of Ukraine using a modified TSPLIB library and the DEAP platform created in the Python programming language. A block diagram of the GA is presented, the main elements of which are the tournament selection operator, the ordered crossover operator, and the inversion mutation operator. The influence of the population size and the number of generations on the asymptotic complexity of the genetic algorithm in solving the TSP problem is studied. The study considered changing the size of the GA population from 50 to 500 with a step of 50, while for each such value four sets of the number of generations were modelled: from 50 to 200 with a step of 50. Based on the obtained results, we show a linear dependence of the GA execution time on the size of the considered input data. It is shown that the smallest time complexity of the presented GA for the given TSP problem is 0.33848 seconds with a population size of 50 and a similar number of generations, while the largest value is 3.752734 seconds with a population size of 500 and a number of generations of 200. The obtained results can be used to optimise the performance of a GA in the TSP problem.

COMPARATIVE ANALYSIS OF THE APPLICATION OF HEURISTIC ALGORITHMS FOR SOLVING THE TSP PROBLEM

The need to solve the traveling salesman problem (TSP) often arises when solving practically significant optimization problems, such as problems in the field of economics, logistics in the widest range of applications, in chains of technical programs. Quite often, the specifics of these problems require obtaining a solution that is as close to the exact value as possible. But the TSP problem is NP-complex, that is, its exact solution can be obtained only in exponential time. Therefore, it is not efficient to solve the TSP problem by the full search algorithm in the presence of a large number of vertices of the graph. However, there are various heuristic algorithms that allow finding a rational solution to this problem with a large number of vertices in a time acceptable for the relevant subject area. In this work, the problem of the traveling salesman is defined as a mathematical programming task of finding the shortest path for the movement of a traveling salesman (traveling salesman), the goal of which is to visit all the objects involved in the task in the shortest time and with minimal costs. Appropriate adaptations of the heuristic algorithms, namely the genetic algorithm and the ant colony algorithm, were developed in the MATLAB environment. A computational experiment was performed on the same input sample, a comparative analysis of the performance of two heuristic algorithms, and the effectiveness of the use of heuristic algorithms for solving NP-complex problems was proven.

Traveling of multiple salesmen to dynamically changing locations for satisfying multiple goals

Polymer grade scheduling, maritime surveillance, e-food delivery, e-commerce, and military tactics necessitate multiple agents (e.g., extruders, speed boats, salesmen) capable of visiting (or completing) dynamically changing locations (or tasks) in minimum time and distance. This study proposes a novel methodology based on clustering and local heuristic-based evolutionary algorithms to address the dynamic traveling salesman problem (TSP) and the dynamic multi-salesman problem with multiple objectives. The proposed algorithm is evaluated on 11 benchmark TSP problems and large-scale problems with up to 10,000 instances. The results show the superior performance of the proposed methodology called the dynamic two-stage evolutionary algorithm as compared to the dynamic hybrid local search evolutionary algorithm. Furthermore, the algorithm's applicability is illustrated through various scenarios involving up to four salesmen and three objectives with dynamically changing locations. To demonstrate real-world relevance, a maritime surveillance problem employing a helideck monitoring system is solved, wherein the objective is to minimize the patrolling route while visiting faulty vessels that threaten marine vessels. This study provides a general framework of TSP which finds application in several sectors, including planning and scheduling in chemical and manufacturing industries, the defense sector, and the e-commerce sector. Finally, the results showcase the effectiveness of the proposed methodology in solving the dynamic multiobjective, and multiple salesmen problem, which represents a more generalized version of the TSP.

Application of ant algorithm in solving TSP problems

Ant colony system algorithms, as a new class of global search algorithms, can solve the TSP (Traveling Salesman Problem). At the same time, the TSP problem is a classical NP-C problem, the solution of TSP problem involves many fields, such as network routing, vehicle routing, logistics and transportation, so it is very important to solve TSP problem effectively. In this experiment, the ant colony optimization algorithm is simulated and experimented using Matlab software on bayg29 dataset and the algorithm strategy is improved by controlling different parameters. Our main goal is to improve the performance of the algorithm through meticulous parameter fine-tuning and policy improvement. We explored a range of parameter adjustments, including variables such as pheromone evaporation rate, ant colony size, etc. By undertaking this comprehensive investigation, we aspire to push the boundaries of ant colony optimization, unlocking its potential to deliver highly efficient solutions for complex TSP instances. This research holds promise for revolutionizing the efficiency and cost-effectiveness of solutions across a wide range of practical applications.

LEADERS AND FOLLOWERS ALGORITHM FOR TRAVELING SALESMAN PROBLEM

Leaders and Followers algorithm is a metaheuristics algorithm. In solving continuous optimization, this algorithm is proved to be better than other well-known algorithms, such as Genetic Algorithm and Particle Swarm Optimization. This paper aims to apply the Leaders and Followers algorithm for the Traveling Salesman Problem (TSP), a well-known combinatorial optimization problem to minimize distance. There are some modifications in order to fit the algorithm in TSP problems. Some most-used-problems in TSP are used to test this algorithm. The result is that the Leaders and Followers algorithm performs well, stable, and guarantees the optimality of the obtained solution in TSP with fewer than 20 cities. In TSP with a bigger number of cities, the proposed algorithm is not stable and might has difficulties in finding the optimal solutions.

Correcting misinterpretations on the distribution of feasible solution lengths in the traveling salesman problem

The traveling salesman problem (TSP) is the canonical combinatorial optimization problem famous throughout literature. There exists an objective function associated with every feasible solution. However, the increase in the number of possible solutions makes this an NP-Hard problem. We show that the central limit theorem (CLT) applies to the problem. We then conduct extensive computational testing to show that the cycle lengths tend to a normal distribution as the problem grows large. When the size of the TSP problem exceeds computational power, better understanding solution distributions allows us to save resources. This is a non-trivial result as understanding solution distributions in huge TSP problems helps us to minimize computational effort that may not lead to significantly better results.

G-DGANet: Gated deep graph attention network with reinforcement learning for solving traveling salesman problem

Combinatorial optimization problem (COP) is an NP-hard problem for which finding an optimal solution is difficult, especially as the problem size increases. The Traveling Salesman Problem (TSP), one of the COPs that can be formulated over a graph, is a well-researched area in operations research and computer science. Deep Reinforcement Learning (DRL) is now regarded as a promising approach for solving TSP and other NP hard problems. In this paper, we propose a novel Gated Deep Graph Attention Network (G-DGANet) which builds upon the existing Graph Neural Network (GNN) to solve TSP. The proposed G-DGANet uses gating mechanism between subsequent layers of the network to extract representations of nodes deeper in the network without loss in performance. G-DGANet also designs a novel aggregator to construct global graph embeddings from different embedding preferences. In addition, to effectively learn underlying structure of a graph, G-DGANet integrates node and edge information of the graph while updating node representations in the message passing mechanism. We used proximal policy optimization (PPO) to train G-DGANet on randomly generated instances. We conducted an experiment on randomly generated instances and on real-world road network data generated from digital maps to verify the performance of G-DGANet. The findings from experiments demonstrate that G-DGANet outperforms most traditional heuristics and existing DRL approaches, with high generalization abilities from random instance training to random instance testing and real-world road network instance testing.

CSGA: A DUAL POPULATION GENETIC ALGORITHM BASED ON MEXICAN CAVEFISH GENETIC DIVERSITY

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.

Hybrid iterated local search algorithm for optimization route of airplane travel plans

<span lang="EN-US">The traveling salesman problem (TSP) is a very popular combinatorics problem. This problem has been widely applied to various real problems. The TSP problem has been classified as a Non-deterministic Polynomial Hard (NP-Hard), so a non-deterministic algorithm is needed to solve this problem. However, a non-deterministic algorithm can only produce a fairly good solution but does not guarantee an optimal solution. Therefore, there are still opportunities to develop new algorithms with better optimization results. This research develops a new algorithm by hybridizing three local search algorithms, namely, iterated local search (ILS) with simulated annealing (SA) and hill climbing (HC), to get a better optimization result. This algorithm aimed to solve TSP problems in the transportation sector, using a case study from the Traveling Salesman Challenge 2.0 (TSC 2.0). The test results show that the developed algorithm can optimize better by 15.7% on average and 11.4% based on the best results compared to previous studies using the Tabu-SA algorithm.</span>

An Improvement to the 2-Opt Heuristic Algorithm for Approximation of Optimal TSP Tour

The travelling salesman problem (TSP) is perhaps the most researched problem in the field of Computer Science and Operations. It is a known NP-hard problem and has significant practical applications in a variety of areas, such as logistics, planning, and scheduling. Route optimisation not only improves the overall profitability of a logistic centre but also reduces greenhouse gas emissions by minimising the distance travelled. In this article, we propose a simple and improved heuristic algorithm named 2-Opt++, which solves symmetric TSP problems using an enhanced 2-Opt local search technique, to generate better results. As with 2-Opt, our proposed method can also be applied to the Vehicle Routing Problem (VRP), with minor modifications. We have compared our technique with six existing algorithms, namely ruin and recreate, nearest neighbour, genetic algorithm, simulated annealing, Tabu search, and ant colony optimisation. Furthermore, to allow for the complexity of larger TSP instances, we have used a graph compression/candidate list technique that helps in reducing the computational complexity and time. The comprehensive empirical evaluation carried out for this research work shows the efficacy of the 2-Opt++ algorithm as it outperforms the other well-known algorithms in terms of the error margin, execution time, and time of convergence.

A Labelling Method for the Travelling Salesman Problem

The travelling salesman problem (TSP) is a problem whereby a finite number of nodes are supposed to be visited exactly once, one after the other, in such a way that the total weight of connecting arcs used to visit these nodes is minimized. We propose a labelling method to solve the TSP problem. The algorithm terminates after K−1 iterations, where K is the total number of nodes in the network. The algorithm’s design allows it to determine alternative tours if there are any in the TSP network. The computational complexity of the algorithm reduces as iterations increase, thereby making it a powerful and efficient algorithm. Numerical illustrations are used to prove the efficiency and validity of the proposed algorithm.

Using Self-organizing Maps to Solve the Travelling Salesman Problem: A Review

This survey paper presents a collection of the most important algorithms for the well-known Traveling Salesman Problem (TSP) using Self-Organizing Maps (SOM). Each one of the presented models is characterized by its own features and advantages. The modes are compared to each other to find their differences and similarities. The models are classified in two basic categories, namely the enriched and hybrid models. For each model we present information regarding its performance, the required number of iterations, as well as the number of neurons that are capable of solving the TSP problem. Based on the experimental results, the best model is identified for different occasions. The paper is a good starting point for anyone who is interested in solving TSP with SOM and desires to grasp a lot about this renowned problem.

Memory-efficient Transformer-based network model for Traveling Salesman Problem

Combinatorial optimization problems such as Traveling Salesman Problem (TSP) have a wide range of real-world applications in transportation, logistics, manufacturing. It has always been a difficult problem to solve large-scale TSP problems quickly because of memory usage limitations. Recent research shows that the Transformer model is a promising approach. However, the Transformer has several severe problems that prevent it from quickly solving TSP combinatorial optimization problems, such as quadratic time complexity, especially quadratic space complexity, and the inherent limitations of the encoder and decoder itself. To address these issues, we developed a memory-efficient Transformer-based network model for TSP combinatorial optimization problems, termed Tspformer, with two distinctive characteristics: (1) a sampled scaled dot-product attention mechanism with O(Llog(L)) (L is the length of input sequences) time and space complexity, which is the most different between our work and other works. (2) due to the reduced space complexity, GPU/CPU memory usage is significantly reduced. Extensive experiments demonstrate that Tspformer significantly outperforms existing methods and provides a new solution to the TSP combinatorial optimization problems. Our Pytorch code will be publicly available on GitHub https://github.com/yhnju/tspFormer.

Розв’язання задачі комівояжера на основі методу відпалу з урахуванням нечіткості сприйняття плину часу

This paper investigates the use of fuzzy numbers and the annealing method to improve the results of the traveling salesman problem (TSP) by more accurately representing real-world circumstances, where the value of the objective function represents the subjective perception of the length of the time interval required to travel between cities. TSP is a classic combinatorial optimization problem that involves finding the shortest route between a set of cities. Fuzzy numbers are used to model input inaccuracy and uncertainty, as they allow for a more detailed representation of real-world constraints and factors that may affect the problem. The annealing method is used to optimize the TSP solution by gradually decreasing the temperature of the system, which allows exploring different solutions and avoiding getting stuck in local minima. To demonstrate the effectiveness of this approach, a Python program was developed that was used to compare the results of the TSP problem using crisp and fuzzy numbers using the annealing method. The results show that the use of fuzzy numbers, particularly triangular and parabolic, with the annealing method leads to a significant improvement in the results of the TSP problem compared to the use of crisp numbers, assuming a model is called realistic if it has possible deviations from the expected fixed mean. Computational results of the program are presented and analyzed, demonstrating the potential of this approach for real-world optimization problems involving imprecise or uncertain data and which can be particularly applied to the optimization of processes with subjective time perception.

Experimental Evaluation of Genetic Algorithms to Solve the DNA Assembly Optimization Problem

This paper aims to highlight the motivations for investigating genetic algorithms (GAs) to solve the DNA Fragment Assembly (DNAFA) problem. DNAFA problem is an optimization problem that attempts to reconstruct an original DNA sequence by finding the shortest DNA sequence from a given set of fragments. This paper is a continuation of our previous research paper in which the existence of a polynomial-time reduction of DNAFA into the Traveling Salesman Problem (TSP) and the Quadratic Assignment Problem (QAP) was discussed. Taking advantage of this reduction, this work conceptually designed a genetic algorithm (GA) platform to solve the DNAFA problem. This platform offers several ingredients enabling us to create several variants of GA solvers for the DNAFA optimization problems. The main contribution of this paper is the designing of an efficient GA variant by carefully integrating different GAs operators of the platform. For that, this work individually studied the effects of different GAs operators on the performance of solving the DNAFA problem. This study has the advantage of benefiting from prior knowledge of the performance of these operators in the contexts of the TSP and QAP problems. The best designed GA variant shows a significant improvement in accuracy (overlap score) reaching more than 172% of what is reported in the literature.

Solving TSP Problems with Integer Programming

The traveling salesman problem (TSP) is one of the NP-hard problems in combinatorial optimization. This paper focuses on finding the optimal planning path using integer programming and adding time allocation to minimize the cost. The use and accuracy of various algorithms are also compared in solving this problem.

Load Balancing Based on Firefly and Ant Colony Optimization Algorithms for Parallel Computing.

With the wide application of computational fluid dynamics in various fields and the continuous growth of the complexity of the problem and the scale of the computational grid, large-scale parallel computing came into being and became an indispensable means to solve this problem. In the numerical simulation of multi-block grids, the mapping strategy from grid block to processor is an important factor affecting the efficiency of load balancing and communication overhead. The multi-level graph partitioning algorithm is an important algorithm that introduces graph network dynamic programming to solve the load-balancing problem. This paper proposed a firefly-ant compound optimization (FaCO) algorithm for the weighted fusion of two optimization rules of the firefly and ant colony algorithm. For the graph, results after multi-level graph partitioning are transformed into a traveling salesman problem (TSP). This algorithm is used to optimize the load distribution of the solution, and finally, the rough graph segmentation is projected to obtain the most original segmentation optimization results. Although firefly algorithm (FA) and ant colony optimization (ACO), as swarm intelligence algorithms, are widely used to solve TSP problems, for the problems for which swarm intelligence algorithms easily fall into local optimization and low search accuracy, the improvement of the FaCO algorithm adjusts the weight of iterative location selection and updates the location. Experimental results on publicly available datasets such as the Oliver30 dataset and the eil51 dataset demonstrated the effectiveness of the FaCO algorithm. It is also significantly better than the commonly used firefly algorithm and other algorithms in terms of the search results and efficiency and achieves better results in optimizing the load-balancing problem of parallel computing.

A Novel Discrete Group Teaching Optimization Algorithm for TSP Path Planning with Unmanned Surface Vehicles

A growing number of researchers are interested in deploying unmanned surface vehicles (USVs) in support of ocean environmental monitoring. To accomplish these missions efficiently, multiple-waypoint path planning strategies for survey USVs are still a key challenge. The multiple-waypoint path planning problem, mathematically equivalent to the traveling salesman problem (TSP), is addressed in this paper using a discrete group teaching optimization algorithm (DGTOA). Generally, the algorithm consists of three phases. In the initialization phase, the DGTOA generates the initial sequence for students through greedy initialization. In the crossover phase, a new greedy crossover algorithm is introduced to increase diversity. In the mutation phase, to balance the exploration and exploitation, this paper proposes a dynamic adaptive neighborhood radius based on triangular probability selection to apply in the shift mutation algorithm, the inversion mutation algorithm, and the 3-opt mutation algorithm. To verify the performance of the DGTOA, fifteen benchmark cases from TSPLIB are implemented to compare the DGTOA with the discrete tree seed algorithm, discrete Jaya algorithm, artificial bee colony optimization, particle swarm optimization-ant colony optimization, and discrete shuffled frog-leaping algorithm. The results demonstrate that the DGTOA is a robust and competitive algorithm, especially for large-scale TSP problems. Meanwhile, the USV simulation results indicate that the DGTOA performs well in terms of exploration and exploitation.

Simulated Annealing Algorithm for Determining Travelling Salesman Problem Solution and Its Comparison with Branch and Bound Method

Travelling Salesman Problem (TSP) is a problem where a person must visit some places, starting from one city and then moving on to the next city with the conditions that the places visited can only be passed precisely once and then back to the starting city. TSP is an NP-hard, an important problem in operations research. TSP problems can be solved by an exact method or an approximation method, namely the metaheuristic method. This research aims to solve the TSP problem with an approximation method called the Simulated Annealing (SA), and then compare the results of this approximation method with the exact Branch and Bound method. The results indicated that the SA method could accomplish TSP problems. However, like other metaheuristic methods, SA only accomplishes it using an approach to get good results. Still, it cannot be determined that SA has the most optimal results, but the time needed by the SA method is faster than the Branch and Bound method. In case I, the percentage difference between the distance generated using the SA method with the B-and-B method is 0%, in case II it is 7% and in case III it is 8%.