Salesman Problem Research Articles

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.

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.

An Analysis of Hierarchical Routing Strategy with Advanced Additional Sensors in WSNs

Two kinds of sensors will be discussed in some sort of wireless sensor networks (WSNs). One is named a normal sensor (called A_nodes) with fixed initial energy, which can get perception data from the surrounding environment and have functions of storage and forwarding. The other is named relay sensor (called B_node) with sufficient energy, which only can store data and forward data. Cluster heads (called A_heads) are chosen among A_nodes by probability mode to create clusters first. Local adjustments would be done inter-cluster. Then the sink is selected as the root and a backbone shortest path tree with hops limited is built dynamically from A_heads, B_Nodes and sink. Inside the backbone shortest path aggregation tree, two steps are done. The first step, select some B_node as cluster head (denoted as B_node) and make local adjustment intra-clusters if it is possible. The second step, adjust the shortest path aggregation tree. By using both particle swarm optimization (PSO) and ant colony optimization (ACO), a traveling salesman problem (TSP) cycle with shorter path length is obtained (this method is noted as TSP_PSO_ACO). This TSP cycle consists of all nodes in the aggregation backbone tree. Along this cycle, energy is offered to nodes except sink. The above strategy would be discussed for its feasibility and time complexity. The experimental results imply that the energy consumption is reduced by using local adjustment. The relative failure rate of nodes can be reduced in the later stage of WSN survival. Offering energy to a small number of nodes several times has no obvious impact on the overall energy consumption and survival time of WSN. The case with some small-sized obstacles in WSN area is also discussed. There are little affections on TSP cycles both from results on theoretical analysis and simulation if there are some obstacles with small size in WSN area.

Multi-start team orienteering problem for UAS mission re-planning with data-efficient deep reinforcement learning

In this paper, we study the Multi-Start Team Orienteering Problem (MSTOP), a mission re-planning problem where vehicles are initially located away from the depot and have different amounts of fuel. We consider/assume the goal of multiple vehicles is to travel to maximize the sum of collected profits under resource (e.g., time, fuel) consumption constraints. Such re-planning problems occur in a wide range of intelligent UAS applications where changes in the mission environment force the operation of multiple vehicles to change from the original plan. To solve this problem with deep reinforcement learning (RL), we develop a policy network with self-attention on each partial tour and encoder-decoder attention between the partial tour and the remaining nodes. We propose a modified REINFORCE algorithm where the greedy rollout baseline is replaced by a local mini-batch baseline based on multiple, possibly non-duplicate sample rollouts. By drawing multiple samples per training instance, we can learn faster and obtain a stable policy gradient estimator with significantly fewer instances. The proposed training algorithm outperforms the conventional greedy rollout baseline, even when combined with the maximum entropy objective. The efficiency of our method is further demonstrated in two classical problems – the Traveling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP). The experimental results show that our method enables models to develop more effective heuristics and performs competitively with the state-of-the-art deep reinforcement learning methods.

Solution of Uncertain Constrained Multi-Objective Travelling Salesman Problem with Aspiration Level Based Multi Objective Quasi Oppositional Jaya Algorithm

Solution of Uncertain Constrained Multi-Objective Travelling Salesman Problem with Aspiration Level Based Multi Objective Quasi Oppositional Jaya Algorithm

Multiobjective Combinatorial Optimization Using a Single Deep Reinforcement Learning Model.

This article proposes utilizing a single deep reinforcement learning model to solve combinatorial multiobjective optimization problems. We use the well-known multiobjective traveling salesman problem (MOTSP) as an example. Our proposed method employs an encoder-decoder framework to learn the mapping from the MOTSP instance to its Pareto-optimal set. Specifically, it leverages a novel routing encoder to extract information for both the entire multiobjective aspect and every individual objective from the MOTSP instance. The global embeddings and each objective's embeddings are adaptively aggregated via a routing network to form the subproblems' embedding that can well represent the MOTSP features. Using a modified context embedding, the subproblems' embeddings are fed into a decoder to produce a set of approximate Pareto-optimal solutions in parallel. Additionally, we develop a Top-k baseline to enable more efficient data utilization and lightweight training for our proposed method. We compare our method with heuristic-based and learning-based ones on various types of MOTSP instances, and the experimental results show that our method can solve MOTSP instances in real-time and outperform the other algorithms, especially on large-scale problem instances.

Modeling and Solving the Traveling Salesman Problem with Speed Optimization for a Plug-In Hybrid Electric Vehicle

This paper investigates a variant of the traveling salesman problem (TSP) with speed optimization for a plug-in hybrid electric vehicle (PHEV), simultaneously optimizing the average speed and operation mode for each road segment in the route. Two mixed-integer nonlinear programming models are proposed for the problem: one with continuous speed decision variables and one with discretized variables. Because the models are nonlinear, we propose reformulation schemes and introduce valid inequalities to strengthen them. We also describe a branch-and-cut algorithm to solve these reformulations. Extensive numerical experiments are performed to demonstrate the algorithm’s performance in terms of computing time and energy consumption costs. Specifically, the proposed solution method can efficiently solve instances with a realistic number of customers and outperforms the benchmark approaches from the literature. Integrating speed optimization into the TSP of a PHEV can lead to significant energy savings compared with the fixed-speed TSP. In addition, the proposed model is extended to investigate the impact of the presence of charging stations, which makes the problem harder to solve but has the potential to further reduce energy consumption costs. Funding: F. Wu gratefully acknowledges the support of the National Natural Science Foundation of China [Grant 72271161]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0247 .

Enhancing the Dhouib-Matrix-4 metaheuristic to generate the Pareto non-dominated set solutions for multi-objective travelling salesman problem: The DM4-PMO method

This paper focuses on the multi-objective Travelling Salesman Problem for which the aim is to find the set of efficient solutions. To obtain this Pareto set's solutions, a novel metaheuristic named DM4-PMO is proposed. The DM4-PMO is based on the first enhancement of the new Dhouib-Matrix-4 (DM4) method and composed of two steps. First, a weighted sum function with a multi variation of weights is used to find the first non-dominated pareto frontier solutions. Second, a lexicographical resolution is applied in some non-dominated solutions found in the first Pareto frontier to generate the final Pareto frontier solutions. The performance of the proposed approach is demonstrated by the experiment on two-objective problems that are taken from TSP-LIB and DIMACS datasets. The test results show that the proposed DM4-PMO is robust, fast, and simply structured, and obtain a Pareto non-dominated set solutions in short computational times using very few user-defined parameters.

Research on dual robot collaboration method based on improved double ant colony algorithm

Purpose This study aims to propose an efficient dual-robot task collaboration strategy to address the issue of low work efficiency and inability to meet the production needs of a single robot during construction operations. Design/methodology/approach A hybrid task allocation method based on integer programming and auction algorithms, with the aim of achieving a balanced workload between two robots has been proposed. In addition, while ensuring reasonable workload allocation between the two robots, an improved dual ant colony algorithm was used to solve the dual traveling salesman problem, and the global path planning of the two robots was determined, resulting in an efficient and collision-free path for the dual robots to operate. Meanwhile, an improved fast Random tree rapidly-exploring random tree algorithm is introduced as a local obstacle avoidance strategy. Findings The proposed method combines randomization and iteration techniques to achieve an efficient task allocation strategy for two robots, ensuring the relative optimal global path of the two robots in cooperation and solving complex local obstacle avoidance problems. Originality/value This method is applied to the scene of steel bar tying in construction work, with the workload allocation and collaborative work between two robots as evaluation indicators. The experimental results show that this method can efficiently complete the steel bar banding operation, effectively reduce the interference between the two robots and minimize the interference of obstacles in the environment.

Sustainable multiple travelling salesman problem solved by reference point aspiration level based multi objective quasi oppositional Jaya algorithm under uncertain environment

Sustainable multiple travelling salesman problem solved by reference point aspiration level based multi objective quasi oppositional Jaya algorithm under uncertain environment

Complexity of Some Types of Cyclic Snake Graphs

The number of spanning trees in graphs (networks) is a crucial invariant, and it is also an important measure of the reliability of a network. Spanning trees are special subgraphs of a graph that have several important properties. First, <I>T</I> must span <I>G</I>, which means it must contain every vertex in graph <I>G</I>, if <I>T</I> is a spanning tree of graph <I>G</I>. <I>T</I> needs to be a subgraph of <I>G</I>, second. Stated differently, any edge present in <I>T</I> needs to be present in <I>G </I>as well. Third, <I>G</I> is the same as <I>T</I> if each edge in <I>T</I> is likewise present in <I>G</I>. In path-finding algorithms like Dijkstra's shortest path algorithm and A* search algorithm, spanning trees play an essential part. In those approaches, spanning trees are computed as component components. Protocols for network routing also take advantage of it. In numerous techniques and applications, minimum spanning trees are highly beneficial. Computer networks, electrical grids, and water networks all frequently use them. They are also utilized in significant algorithms like the min-cut max-flow algorithm and in graph issues like the travelling salesperson problem. In this paper, we use matrix analysis and linear algebra techniques to obtain simple formulas for the number of spanning trees of certain kinds of cyclic snake graphs.

Transfer Reinforcement Learning for Combinatorial Optimization Problems

Reinforcement learning is an important technique in various fields, particularly in automated machine learning for reinforcement learning (AutoRL). The integration of transfer learning (TL) with AutoRL in combinatorial optimization is an area that requires further research. This paper employs both AutoRL and TL to effectively tackle combinatorial optimization challenges, specifically the asymmetric traveling salesman problem (ATSP) and the sequential ordering problem (SOP). A statistical analysis was conducted to assess the impact of TL on the aforementioned problems. Furthermore, the Auto_TL_RL algorithm was introduced as a novel contribution, combining the AutoRL and TL methodologies. Empirical findings strongly support the effectiveness of this integration, resulting in solutions that were significantly more efficient than conventional techniques, with an 85.7% improvement in the preliminary analysis results. Additionally, the computational time was reduced in 13 instances (i.e., in 92.8% of the simulated problems). The TL-integrated model outperformed the optimal benchmarks, demonstrating its superior convergence. The Auto_TL_RL algorithm design allows for smooth transitions between the ATSP and SOP domains. In a comprehensive evaluation, Auto_TL_RL significantly outperformed traditional methodologies in 78% of the instances analyzed.

A New Approach Based on Collective Intelligence to Solve Traveling Salesman Problems.

This paper presents a novel approach based on the ant system algorithm for solving discrete optimization problems. The proposed method is based on path construction, path improvement techniques, and the footprint mechanism. Some information about the optimization problem and collective intelligence is used in order to create solutions in the path construction phase. In the path improvement phase, neighborhood operations are applied to the solution, which is the best of the population and is obtained from the path construction phase. The collective intelligence in the path construction phase is based on a footprint mechanism, and more footprints on the arc improve the selection chance of this arc. A selection probability is also balanced by using information about the problem (e.g., the distance between nodes for a traveling salesman problem). The performance of the proposed method has been investigated on 25 traveling salesman problems and compared with state-of-the-art algorithms. The experimental comparisons show that the proposed method produced comparable results for the problems dealt with in this study.

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.

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.

Hybrid Arithmetic-Rider Optimization algorithm as new intelligent model for travelling salesman problem

Hybrid Arithmetic-Rider Optimization algorithm as new intelligent model for travelling salesman problem

Solving the traveling salesman problem using reinforcement learning

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.

A Twin Learning Framework for Traveling Salesman Problem Based on Autoencoder, Graph Filter, and Transfer Learning

Efficient solvers for traveling salesman problem (TSP) have great significance in the field of consumer electronic systems and devices. Existing studies require independent and repetitive runs for similar TSPs. To utilize useful knowledge buried in the twin TSP which is similar to the target TSP, a twin learning framework based on task matching and mapping strategy is proposed. We use an autoencoder to extract the feature vectors of the historical tasks and the target task to find the twin task. If not found, construct a twin task based on a graph-filter. Further, we get the solution of the target task with the optimized twin task by learning a mapping matrix from the twin task to the target task. Finally, we incorporate local search algorithms into the twin learning framework, which called strategy mapping solver (SMS) to further improve the quality of the solution. The efficacy of the SMS is evaluated through comprehensive empirical studies with commonly used TSP benchmarks, against meta-heuristic algorithms and a learning improvement heuristics algorithm, demonstrating its effectiveness for TSP. Moreover, a real-world combinatorial optimization application, laser engraving path planning, is presented to further confirm the efficacy of SMS.

A Knowledge-Guided End-to-End Optimization Framework Based on Reinforcement Learning for Flow Shop Scheduling

Designing an effective and efficient end-to-end optimization framework with good generalization for shop scheduling is an emerging topic in the informational manufacturing system. Existing end-to-end frameworks have achieved satisfactory results for COPs such as traveling salesman problem and vehicle routing problem. However, the performances of these methods in solving complex COPs such as shop scheduling need to be improved. In this paper, a knowledge-guided end-to-end optimization framework based on reinforcement learning is proposed to solve the permutation flow shop scheduling problem (PFSP). Firstly, a new policy network is designed based on the problem characteristics to deal with different scales of PFSPs and achieve iterative end-to-end generation. Secondly, an improved policy-based reinforcement learning algorithm by using the knowledge accumulated during the training process is designed to enhance the training quality. Thirdly, a knowledge-guided improvement strategy is introduced through the cooperation of local search and supervised learning to improve the learning of the policy. Simulation results and comparisons show that the knowledge-guided end-to-end optimization framework can obtain better results than different kinds of commonly-used optimization methods in limited computation time for solving the PFSP.

Decremental State-Space Relaxations for the Basic Traveling Salesman Problem with a Drone

Truck-and-drone routing problems have become an important research topic in the last decade because of their applications for last-mile deliveries. Despite the many publications in this area, the most efficient exact algorithms designed thus far struggle to solve the benchmark instances with 39 or more customers. This fact holds even for one of the simplest variants involving one truck and one drone whose routes must synchronize at customers’ locations: the basic traveling salesman problem with a drone (TSP-D). In this article, we devise a new algorithm for the TSP-D that solves every benchmark instance with up to 59 customers, and it scales up to 99 customers when the drone is much faster than the truck. The core of our method is a dynamic programming algorithm that is applied for column generation and variable fixing within tailored decremental state-space relaxation strategies. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by Fondo para la Investigación Científica y Tecnológica [Grant PICT-2018-2961] (Ministry of Science, Argentina). Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0390 .