Traveling Salesman Problem Instances Research Articles

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.

An edge-aware graph autoencoder trained on scale-imbalanced data for traveling salesman problems

In recent years, there has been a notable surge in research on machine learning techniques for combinatorial optimization. It has been shown that learning-based methods outperform traditional heuristics and mathematical solvers on the Traveling Salesman Problem (TSP) in terms of both performance and computational efficiency. However, most learning-based TSP solvers are primarily designed for fixed-scale TSP instances, and also require a large number of training samples to achieve optimal performance. To fill this gap, this work proposes a data-driven graph representation learning method for solving TSPs with various numbers of cities. Specifically, we formulate the TSP as a link prediction task and propose an edge-aware graph autoencoder (EdgeGAE) model that can solve TSPs by learning from various-scale samples with an imbalanced distribution. A residual gated encoder is trained to learn latent edge embeddings, followed by an edge-centered decoder to output link predictions in an end-to-end manner. Furthermore, we introduce an active sampling strategy into the training process to improve the model’s generalization capability in large-scale scenarios. To investigate the model’s practical applicability, we generate a scale-imbalanced dataset comprising 50,000 TSP instances ranging from 50 to 500 cities. The experimental results demonstrate that the proposed edge-aware graph autoencoder model achieves a highly competitive performance among state-of-the-art graph learning-based approaches in solving TSPs with various scales, implying its remarkable potential in dealing with practical optimization challenges.

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.

An Adaptive Ant Colony Optimization for Solving Large-Scale Traveling Salesman Problem

The ant colony algorithm faces dimensional catastrophe problems when solving the large-scale traveling salesman problem, which leads to unsatisfactory solution quality and convergence speed. To solve this problem, an adaptive ant colony optimization for large-scale traveling salesman problem (AACO-LST) is proposed. First, AACO-LST improves the state transfer rule to make it adaptively adjust with the population evolution, thus accelerating its convergence speed; then, the 2-opt operator is used to locally optimize the part of better ant paths to further optimize the solution quality of the proposed algorithm. Finally, the constructed adaptive pheromone update rules can significantly improve the search efficiency and prevent the algorithm from falling into local optimal solutions or premature stagnation. The simulation based on 45 traveling salesman problem instances shows that AACO-LST improves the solution quality by 79% compared to the ant colony system (ACS), and in comparison with other algorithms, the PE of AACO-LST is not more than 1% and the Err is not more than 2%, which indicates that AACO-LST can find high-quality solutions with high stability. Finally, the convergence speed of the proposed algorithm was tested. The data shows that the average convergence speed of AACO-LST is more than twice that of the comparison algorithm. The relevant code can be found on our project homepage.

Evolving ensembles of heuristics for the travelling salesman problem

AbstractThe Travelling Salesman Problem (TSP) is a well-known optimisation problem that has been widely studied over the last century. As a result, a variety of exact and approximate algorithms have been proposed in the literature. When it comes to solving large instances in real-time, greedy algorithms guided by priority rules represent the most common approach, being the nearest neighbour (NN) heuristic one of the most popular rules. NN is quite general but it is too simple and so it may not be the best choice in some cases. Alternatively, we may design more sophisticated heuristics considering the particular features of families of instances. To do that, we have to consider problem attributes other than the proximity of the next city to build priority rules. However, this process may not be easy for humans and so it is often addressed by some learning procedure. In this regard, hyper-heuristics as Genetic Programming (GP) stands as one of the most popular approaches. Furthermore, a single heuristic, even being good in average, may not be good for a number of instances of a given set. For this reason, the use of ensembles of heuristics is often a good alternative, which raises the problem of building ensembles from a given set of heuristic rules. In this paper, we study the application of two kinds of ensembles to the TSP. Given a set of TSP instances having similar characteristics, we firstly exploit a GP to build a set of heuristics involving a number of problem attributes, and then we build ensembles combining these heuristics by means of a Genetic Algorithm (GA). The experimental study provided valuable insights into the construction and utilisation of single rules and ensembles. It clearly demonstrated that the performance of ensembles justifies the time invested when compared to using individual heuristics.

Multi-ant colony algorithm based on cooperative game and dynamic path tracking

For the problems of slow convergence and low stability in traditional ant colony algorithm when solving large-scale Traveling Salesman Problem (TSP), a multi-ant colony algorithm based on cooperative game and dynamic path tracking (CDMACA) is proposed. Firstly, a novel heterogeneous multi-population is formed by Ant Colony System (ACS) and Max–Min Ant System (MMAS). Secondly, a cooperative game is introduced. In the isomorphic populations, all the populations cooperate and then the pheromone income will be distributed to the participants based on the contribution. Populations with higher contribution will obtain more income, which can improve the accuracy of the solutions; On the other hand, in the heterogeneous populations, a learning mechanism is introduced to enable heterogeneous populations to learn the optimal solution from each other to further improve the accuracy. Thirdly, a dynamic path tracking mechanism is proposed to reward or punish the public paths with pheromone based on the similarity of the optimal solutions, which can enhance the concentration of pheromone on the optimal path, thus improving the convergence. Finally, when the algorithm stalls, a pheromone balance mechanism is introduced. By dynamically cutting the pheromone on the current optimal path, the probability of the optimal path being chosen will decrease, which can help the algorithm jump out of the local optimum. Through a large number of TSP instances, the proposed algorithm in this paper can further improve the accuracy when solving large-scale TSP compared with other improved algorithms.

Generative inverse reinforcement learning for learning 2-opt heuristics without extrinsic rewards in routing problems

Deep reinforcement learning (DRL) has shown promise in solving challenging combinatorial optimization (CO) problems, such as the traveling salesman problem (TSP) and vehicle routing problem (VRP). However, existing DRL methods rely on manually designed reward functions, which may be inaccurate or unrealistic. Moreover, traditional DRL algorithms suffer from unstable training and sparse reward problems. This paper proposes GIRL (Generative Inverse Reinforcement Learning), a method to learn 2-opt heuristics without explicit extrinsic rewards to address these limitations. GIRL combines generative adversarial networks (GANs) and DRL to learn effective policies and reward functions in a reverse end-to-end fashion, improving generalization capabilities. Furthermore, we introduce a self-attentional policy network tailored for 2-opt heuristics and train the framework using a soft actor-critic algorithm along with a discriminator in the GAN.Extensive experiments on various TSP and VRP instances demonstrate superior performance compared to state-of-the-art methods. Moreover, integrating GANs and DRL enables data-driven reward functions, improving accuracy and realism. Using self-attentional networks and the soft actor-critic algorithm enhances training stability and addresses the sparse reward problem. This work advances reinforcement learning techniques in CO, enabling more accurate and practical optimization methods in real-world applications.

Improved ant colony-based algorithm for solving traveling salesmen problem

The purpose of this research is to examine how well the Improved Memetic Continuous Pheromone-based Ant Colony Optimization (ICMPACO) algorithm solves standard Traveling Salesman Problem (TSP) instances in comparison to the basic Ant Colony Optimization (ACO) algorithm and the Improved Ant Colony Optimization (IACO) algorithm. The findings of the computer simulation show that the ICMPACO algorithm has higher optimization capabilities, particularly for the berlin52, eil51, and dantzig42 examples. This superiority is further emphasized through better mean values in comparison to traditional ACO methods. The study also proposes a novel approach, Genetic Ant Colony Optimization (GACO), which integrates ACO with genetic algorithms. Pheromone data is efficiently employed to direct the choice of genetic operation points and maintain the foundational elements within the genetic algorithm. Performance results for various TSP instances reveal the potential of GACO in generating high-quality solutions. These findings suggest that both ICMPACO and GACO methods hold significant promise in addressing complex optimization challenges and could pave the way for further advancements in enhancing optimization algorithm performance.

Comparative Study of Variations in Quantum Approximate Optimization Algorithms for the Traveling Salesman Problem.

The traveling salesman problem (TSP) is one of the most often-used NP-hard problems in computer science to study the effectiveness of computing models and hardware platforms. In this regard, it is also heavily used as a vehicle to study the feasibility of the quantum computing paradigm for this class of problems. In this paper, we tackle the TSP using the quantum approximate optimization algorithm (QAOA) approach by formulating it as an optimization problem. By adopting an improved qubit encoding strategy and a layer-wise learning optimization protocol, we present numerical results obtained from the gate-based digital quantum simulator, specifically targeting TSP instances with 3, 4, and 5 cities. We focus on the evaluations of three distinctive QAOA mixer designs, considering their performances in terms of numerical accuracy and optimization cost. Notably, we find that a well-balanced QAOA mixer design exhibits more promising potential for gate-based simulators and realistic quantum devices in the long run, an observation further supported by our noise model simulations. Furthermore, we investigate the sensitivity of the simulations to the TSP graph. Overall, our simulation results show that the digital quantum simulation of problem-inspired ansatz is a successful candidate for finding optimal TSP solutions.

H-TSP: Hierarchically Solving the Large-Scale Traveling Salesman Problem

We propose an end-to-end learning framework based on hierarchical reinforcement learning, called H-TSP, for addressing the large-scale Traveling Salesman Problem (TSP). The proposed H-TSP constructs a solution of a TSP instance starting from the scratch relying on two components: the upper-level policy chooses a small subset of nodes (up to 200 in our experiment) from all nodes that are to be traversed, while the lower-level policy takes the chosen nodes as input and outputs a tour connecting them to the existing partial route (initially only containing the depot). After jointly training the upper-level and lower-level policies, our approach can directly generate solutions for the given TSP instances without relying on any time-consuming search procedures. To demonstrate effectiveness of the proposed approach, we have conducted extensive experiments on randomly generated TSP instances with different numbers of nodes. We show that H-TSP can achieve comparable results (gap 3.42% vs. 7.32%) as SOTA search-based approaches, and more importantly, we reduce the time consumption up to two orders of magnitude (3.32s vs. 395.85s). To the best of our knowledge, H-TSP is the first end-to-end deep reinforcement learning approach that can scale to TSP instances of up to 10000 nodes. Although there are still gaps to SOTA results with respect to solution quality, we believe that H-TSP will be useful for practical applications, particularly those that are time-sensitive e.g., on-call routing and ride hailing service.

Pointerformer: Deep Reinforced Multi-Pointer Transformer for the Traveling Salesman Problem

Traveling Salesman Problem (TSP), as a classic routing optimization problem originally arising in the domain of transportation and logistics, has become a critical task in broader domains, such as manufacturing and biology. Recently, Deep Reinforcement Learning (DRL) has been increasingly employed to solve TSP due to its high inference efficiency. Nevertheless, most of existing end-to-end DRL algorithms only perform well on small TSP instances and can hardly generalize to large scale because of the drastically soaring memory consumption and computation time along with the enlarging problem scale. In this paper, we propose a novel end-to-end DRL approach, referred to as Pointerformer, based on multi-pointer Transformer. Particularly, Pointerformer adopts both reversible residual network in the encoder and multi-pointer network in the decoder to effectively contain memory consumption of the encoder-decoder architecture. To further improve the performance of TSP solutions, Pointerformer employs a feature augmentation method to explore the symmetries of TSP at both training and inference stages as well as an enhanced context embedding approach to include more comprehensive context information in the query. Extensive experiments on a randomly generated benchmark and a public benchmark have shown that, while achieving comparative results on most small-scale TSP instances as state-of-the-art DRL approaches do, Pointerformer can also well generalize to large-scale TSPs.

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.

An Adaptive Layered Clustering Framework with Improved Genetic Algorithm for Solving Large-Scale Traveling Salesman Problems

Traveling salesman problems (TSPs) are well-known combinatorial optimization problems, and most existing algorithms are challenging for solving TSPs when their scale is large. To improve the efficiency of solving large-scale TSPs, this work presents a novel adaptive layered clustering framework with improved genetic algorithm (ALC_IGA). The primary idea behind ALC_IGA is to break down a large-scale problem into a series of small-scale problems. First, the k-means and improved genetic algorithm are used to segment the large-scale TSPs layer by layer and generate the initial solution. Then, the developed two phases simplified 2-opt algorithm is applied to further improve the quality of the initial solution. The analysis reveals that the computational complexity of the ALC_IGA is between O(nlogn) and O(n2). The results of numerical experiments on various TSP instances indicate that, in most situations, the ALC_IGA surpasses the compared two-layered and three-layered algorithms in convergence speed, stability, and solution quality. Specifically, with parallelization, the ALC_IGA can solve instances with 2×105 nodes within 0.15 h, 1.4×106 nodes within 1 h, and 2×106 nodes in three dimensions within 1.5 h.

On the generation of metric TSP instances with a large integrality gap by branch-and-cut

This paper introduces a computational method for generating metric Travelling Salesman Problem (TSP) instances having a large integrality gap. The method is based on the solution of an integer programming problem, called IH-OPT, that takes as input a fractional solution of the Subtour Elimination Problem (SEP) on a TSP instance and computes a TSP instance having an integrality gap larger than or equal to the integrality gap of the first instance. The decision variables of IH-OPT are the entries of the TSP cost matrix, and the constraints are defined by the intersection of the metric cone with an exponential number of inequalities, one for each possible TSP tour. Given the very large number of constraints, we have implemented a branch-and-cut algorithm for solving IH-OPT. Then, by sampling cost vectors over the metric polytope and by solving the corresponding SEP, we can generate random fractional vertices of the SEP polytope. If we solve the IH-OPT problem for every sampled vertex using our branch-and-cut algorithm, we can select the generated TSP instance (i.e., cost vector), yielding the longest runtime for Concorde, the state-of-the-art TSP solver. Our computational results show that our method is very effective in producing challenging instances. As a by-product, we release the Hard-TSPLIB, a library of 41 small metric TSP instances which have a large integrality gap and are challenging in terms of runtime for Concorde.

Discrete komodo algorithm for traveling salesman problem

This work presents a Discrete Komodo Algorithm (DKA) to solve the traveling salesman problem (TSP). Inspired by the Komodo Mlipir Algorithm, a swarm intelligence algorithm based on the behavior of komodo dragons, DKA introduces an enhanced edge-construction operator and a novel edge-destruction operator to explore the discrete search space of TSP. To prioritize the exploration of potential edges, DKA uses a Priority Queue that stores all possible edges with their weights, and its counts appear on tour as its priority. The edges with smaller weights and counts have higher probabilities of being selected for the tour. DKA is tested with 45 different TSP instances from TSPLIB. Experimental results show that DKA guarantees produced the best-known solution for small test problems for up to 24 nodes, and the best solutions are within 6.92% of the known optimal solutions. Compared with some of the previous algorithms, DKA outperforms the recent state-of-the-art algorithms and is significantly better than classical metaheuristic algorithms by achieving a lower relative error.

Focused ACO with node relocation procedure for solving large TSP instances

Ant Colony Optimization (ACO) is a well-known family of nature-inspired metaheuristics used to find approximate solutions for challenging optimization problems. Despite its relative simplicity and adaptability to new problem variants, the efficiency of ACO significantly deteriorates with increasing problem size, even though it often provides high-quality solutions. This paper aims to enhance the performance of ACO when solving large instances of the Traveling Salesman Problem (TSP). Specifically, we propose novel improvements to the current state-of-the-art ACO approach, Focused ACO (FACO), that include a node relocation procedure applied during the solution construction process and a greedy heuristic that preserves high-quality solutions from previous algorithm iterations. Our proposed FACO variants demonstrate a substantial performance improvement when solving TSP instances with up to 200k nodes.

Discrete Salp Swarm Algorithm for symmetric traveling salesman problem.

In the Salp Swarm Algorithm (SSA), the update mechanism is inspired by the unique chain movement of the salp swarm. Numerous versions of SSA were already put forward to deal with various optimization problems, but there are very few discrete versions among them. d-opt is improved based on the 2-opt algorithm: a decreasing factor d is introduced to control the range of neighborhood search; TPALS are modified by Problem Aware Local Search (PALS) based on the characteristics of Travelling Salesman Problem (TSP); The second leader mechanism increases the randomness of the algorithm and avoids falling into the local optimal solution to a certain extent. We also select six classical crossover operators to experiment and select Subtour Exchange Crossover (SEC) and the above three mechanisms to integrate them into the SSA algorithm framework to form Discrete Salp Swarm Algorithm (DSSA). In addition, DSSA was tested on 23 known TSP instances to verify its performance. Comparative simulation studies with other advanced algorithms are conducted and from the results, it is observed that DSSA satisfactorily solves TSP.

Heuristic smoothing ant colony optimization with differential information for the traveling salesman problem

The traveling salesman problem (TSP) is an NP complete problem with potential applications. Thus far, numerous approaches have been proposed to solve the TSP including exact, heuristic and metaheuristic methods. Among them, the ant colony optimization (ACO) algorithm, belonging to metaheuristic methods, has proven to be an efficient method for solving the TSP. However, the phenomena of rapid convergence to local optima and unsatisfactory computational accuracy distinctly limit the performance of ACO. This study therefore proposes a novel ACO algorithm (HSDACO) to compensate for these shortcomings. In the HSDACO algorithm, the involvement of heterogeneous population automation at each iteration generates better candidate solutions with maintaining parameter diversity. Then, the implementation of three smoothing techniques with the 2-Opt method guarantees the effective assessment of candidate solutions in favor of higher quality. Next, the introduction of a differential information updating mechanism, using differential edge information obtained from the candidate solutions, enables the evaporation operation of the pheromone trail to obtain more reasonable guidance. Subsequently, the design of evolutionary state estimation and adjustment is used to monitor the population under correct guidance and regulate the evolutionary state effectively and efficiently. Finally, the HSDACO algorithm is evaluated on public TSP benchmark instances and compared with the other state-of-the-art algorithms. Experimental results show that the proposed HSDACO achieves substantial improvement for mid-scale TSP instances and overwhelming advantages for small-scale TSP instances, in terms of solution accuracy and convergence speed.

A hybrid algorithm based on state-adaptive slime mold model and fractional-order ant system for the travelling salesman problem

AbstractThe ant colony optimization (ACO) is one efficient approach for solving the travelling salesman problem (TSP). Here, we propose a hybrid algorithm based on state-adaptive slime mold model and fractional-order ant system (SSMFAS) to address the TSP. The state-adaptive slime mold (SM) model with two targeted auxiliary strategies emphasizes some critical connections and balances the exploration and exploitation ability of SSMFAS. The consideration of fractional-order calculus in the ant system (AS) takes full advantage of the neighboring information. The pheromone update rule of AS is modified to dynamically integrate the flux information of SM. To understand the search behavior of the proposed algorithm, some mathematical proofs of convergence analysis are given. The experimental results validate the efficiency of the hybridization and demonstrate that the proposed algorithm has the competitive ability of finding the better solutions on TSP instances compared with some state-of-the-art algorithms.

Solving a Multi-Conveyance Travelling Salesman Problem using an Ant Colony Optimization Method

Objectives: A well-known NP-complete problem is the travelling salesman problem (TSP). It has numerous engineering and scientific applications. In this article, we have proposed a multi-conveyance TSP where different conveyances are present to travel from one city to another city. This is an extension to classical TSP. In this TSP, the salesman visits all the cities only once during his/her tour, using different conveyances to travel from one city to another. The cost of travelling between cities using various modes of conveyance varies. The objective of this research is to find the minimum cost tour using an ant colony optimization (ACO) based approach by satisfying the constraints of the proposed multi-conveyance TSP. Method: The considered TSP has been solved using a novel ACO technique. The proposed ACO is adopted with the Roulette-wheel selection and “tuning solution” techniques. We have used a few benchmark datasets from TSPLIB to check the effectiveness of the proposed algorithm. The experimental findings for a few benchmarks TSP instances show that almost always, the proposed ACO is able to find a better result. Then, we used some redefined and randomly generated datasets for experiments. The experimental outcomes for various input datasets are also very encouraging. Findings: The goal of the proposed TSP is to find a complete tour with a minimum cost without exceeding the total travel cost and total travel time. Thus, there are two novel constraints in the classical TSP. Novelty: A unique aspect of the proposed research is the use of several conveyance facilities and a fixed total travel time and cost. This is new, as these three factors are integrated into a single TSP model. Keywords: Travelling Salesman Problem; Travel Cost; Travel Time; MultiConveyance TSP; Ant Colony Optimization