We present an application of the tensor network generator-enhanced optimization (TN-GEO) framework to address the traveling salesman problem (TSP), a fundamental combinatorial optimization challenge. Our approach employs a tensor network Born machine based on automatically differentiable matrix product states (MPS) as the generative model, using the Born rule to define probability distributions over candidate solutions. Unlike approaches based on binary encoding, which require $N^2$ variables and penalty terms to enforce valid tour constraints, we adopt a permutation-based formulation with integer variables and use autoregressive sampling with masking to guarantee that every generated sample is a valid tour by construction. We also introduce a $k$-site MPS variant that learns distributions over $k$-grams (consecutive city subsequences) using a sliding window approach, enabling parameter-efficient modeling for larger instances. Experimental validation on TSPLIB benchmark instances with up to 52 cities demonstrates that TN-GEO can outperform classical heuristics including swap and 2-opt hill-climbing. The $k$-site variants, which put more focus on local correlations, show better results compared to the full-MPS case.
Institute for Advanced Studies（高等研究院）；Aeronautics Institute of Technology（航空技术学院）
Using an enhanced Self-Organizing Map method, we provided suboptimal solutions to the Traveling Salesman Problem. Besides, we employed hyperparameter tuning to identify the most critical features in the algorithm. All improvements in the benchmark work brought consistent results and may inspire future efforts to improve this algorithm and apply it to different problems.
In this paper a highly abstracted view on the historical development of Genetic Algorithms for the Traveling Salesman Problem is given. In a meta-data analysis three phases in the development can be distinguished. First exponential growth in interest till 1996 can be observed, growth stays linear till 2011 and after that publications deteriorate. These three phases are examined and the major milestones are presented. Lastly an outlook to future work in this field is infered.
United Technologies Research Center（联合技术研究中心）；University of Paderborn（帕德博恩大学）
Structure learning of Bayesian networks is an important problem that arises in numerous machine learning applications. In this work, we present a novel approach for learning the structure of Bayesian networks using the solution of an appropriately constructed traveling salesman problem. In our approach, one computes an optimal ordering (partially ordered set) of random variables using methods for the traveling salesman problem. This ordering significantly reduces the search space for the subsequent greedy optimization that computes the final structure of the Bayesian network. We demonstrate ou…
We present a self-learning approach that combines deep reinforcement learning and Monte Carlo tree search to solve the traveling salesman problem. The proposed approach has two advantages. First, it adopts deep reinforcement learning to compute the value functions for decision, which removes the need of hand-crafted features and labelled data. Second, it uses Monte Carlo tree search to select the best policy by comparing different value functions, which increases its generalization ability. Experimental results show that the proposed method performs favorably against other methods in small-to-…
The Traveling Salesman Problem is one of the most intensively studied combinatorial optimization problems due both to its range of real-world applications and its computational complexity. When combined with the Set Covering Problem, it raises even more issues related to tractability and scalability. We study a combined Set Covering and Traveling Salesman problem and provide a mixed integer programming formulation to solve the problem. Motivated by applications where the optimal policy needs to be updated on a regular basis and repetitively solving this via MIP can be computationally expensive…
School of Computer Science, Huazhong University of Science and Technology（华中科技大学计算机科学与技术学院）；Institute of Artificial Intelligence, Huazhong University of Science and Technology（华中科技大学人工智能研究院）；MIS, University of Picardie Jules Verne（皮卡第儒勒-凡尔纳大学MIS实验室）
We address the Traveling Salesman Problem (TSP), a famous NP-hard combinatorial optimization problem. And we propose a variable strategy reinforced approach, denoted as VSR-LKH, which combines three reinforcement learning methods (Q-learning, Sarsa and Monte Carlo) with the well-known TSP algorithm, called Lin-Kernighan-Helsgaun (LKH). VSR-LKH replaces the inflexible traversal operation in LKH, and lets the program learn to make choice at each search step by reinforcement learning. Experimental results on 111 TSP benchmarks from the TSPLIB with up to 85,900 cities demonstrate the excellent per…
Nanyang Technological University（南洋理工大学）；Singapore Institute of Manufacturing Technology（新加坡制造技术研究院）
We present NeuroLKH, a novel algorithm that combines deep learning with the strong traditional heuristic Lin-Kernighan-Helsgaun (LKH) for solving Traveling Salesman Problem. Specifically, we train a Sparse Graph Network (SGN) with supervised learning for edge scores and unsupervised learning for node penalties, both of which are critical for improving the performance of LKH. Based on the output of SGN, NeuroLKH creates the edge candidate set and transforms edge distances to guide the searching process of LKH. Extensive experiments firmly demonstrate that, by training one model on a wide range …
Word embeddings are one of the most fundamental technologies used in natural language processing. Existing word embeddings are high-dimensional and consume considerable computational resources. In this study, we propose WordTour, unsupervised one-dimensional word embeddings. To achieve the challenging goal, we propose a decomposition of the desiderata of word embeddings into two parts, completeness and soundness, and focus on soundness in this paper. Owing to the single dimensionality, WordTour is extremely efficient and provides a minimal means to handle word embeddings. We experimentally con…
NeuFACO: Neural Focused Ant Colony Optimization for Traveling Salesman Problem
College of Engineering and Computer Science, VinUniversity（工程与计算机科学学院，Vin大学）；School of Information and Communications Technology, Hanoi University of Science and Technology（信息与通信技术学院，河内科学技术大学）；University of Engineering and Technology, Vietnam National University（工程大学，越南国家大学）
School of Computer Science and Engineering（计算机科学与工程学院）；NTU, Singapore（新加坡南洋理工大学）；Department of Mathematics（数学系）；Loyola Marymount University（洛约拉马利蒙特大学）
The Traveling Salesman Problem (TSP) is the most popular and most studied combinatorial problem, starting with von Neumann in 1951. It has driven the discovery of several optimization techniques such as cutting planes, branch-and-bound, local search, Lagrangian relaxation, and simulated annealing. The last five years have seen the emergence of promising techniques where (graph) neural networks have been capable to learn new combinatorial algorithms. The main question is whether deep learning can learn better heuristics from data, i.e. replacing human-engineered heuristics? This is appealing be…
Grad. School of Info. Science；Technology Hokkaido University Sapporo, Japan \ .；Technology Hokkaido University Sapporo, Japan；Information Initiative Center Hokkaido University Sapporo, Japan
In this paper, we propose a two-stage optimization strategy for solving the Large-scale Traveling Salesman Problems (LSTSPs) named CCPNRL-GA. First, we hypothesize that the participation of a well-performed individual as an elite can accelerate the convergence of optimization. Based on this hypothesis, in the first stage, we cluster the cities and decompose the LSTSPs into multiple subcomponents, and each subcomponent is optimized with a reusable Pointer Network (PtrNet). After subcomponents optimization, we combine all sub-tours to form a valid solution, this solution joins the second stage o…
Recent works using deep learning to solve the Traveling Salesman Problem (TSP) have focused on learning construction heuristics. Such approaches find TSP solutions of good quality but require additional procedures such as beam search and sampling to improve solutions and achieve state-of-the-art performance. However, few studies have focused on improvement heuristics, where a given solution is improved until reaching a near-optimal one. In this work, we propose to learn a local search heuristic based on 2-opt operators via deep reinforcement learning. We propose a policy gradient algorithm to …
