[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85124-en":3,"doc-seo-85124-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85124,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","GES-TSP: Graph Edge Sparsification for TSP","Solving large-scale Traveling Salesman Problem (TSP) instances exactly is computationally expensive, and conventional graph sparsification often uses fixed heuristics that do not adapt to instance-specific structure. This paper introduces Graph Edge Sparsification (GES), a learning-based method for Euclidean TSP that integrates geometric structural information with combinatorial optimization. GES adaptively generates a sparsification graph per instance, pruning up to 95% of edges on MATILDA with a solution gap under 1%, and reaching over 99% pruning on TSPLIB while keeping the gap below 1%.","GES-TSP: Graph Edge Sparsification for TSP  \nTianfeng Chen  \nSchool of Mathematics and Statistics Lanzhou Universtiy [chentf2025@lzu.edu.cn](chentf2025@lzu.edu.cn)  \nXianyue Li*  \nSchool of Mathematics and Statistics Lanzhou Universtiy [lixianyue@lzu.edu.cn](lixianyue@lzu.edu.cn)  \narXiv :2607 .09708v 1 [ cs .AI] 23 Jun 2026  \nAbstract  \nSolving large-scale instances of the Traveling Salesman Problem (TSP) exactly is computationally expensive. Researchers often employ graph sparsification methods to improve computational efficiency. Traditional sparsification methods typically rely on fixed heuristics and fail to fully exploit instance-specific structural information. In this paper, we propose Graph Edge Sparsification (GES), a learning-based sparsification approach for Euclidean TSP. By incorporating geometric structural information and combinatorial optimization technology, our proposed method adaptively generates a sparsification graph for different instances, significantly reducing the graph size and accelerating the solving process. Experimental results demonstrate that our sparsification method can prune up to 95% of edges on the MATILDA dataset, while keeping the solution gap within 1% of the optimal value.  \nMoreover, our approach exhibits strong generalization capability on the TSPLIB benchmark.In some large-scale instances, the pruning rate exceeds 99%, while the optimality gap remains below 1% .  \n1 Introduction  \nThe Traveling Salesman Problem (TSP) is a classical NP-hard combinatorial optimization problem [1] . Exact methods are computationally expensive and often require lots of time to solve large-scale instances. As a fundamental benchmark problem in combinatorial optimization, the Euclidean TSP has been extensively studied due to its broad applications in transportation planning, circuit design, and routing systems. Improving the efficiency of solving Euclidean TSP instances is therefore of both theoretical and practical significance. In particular, Euclidean TSP instances are typically formulated on complete graphs, where the number of edges grows quadratically with the number of nodes, leading to significant computational overhead in exact or high-quality approximate solvers.  \nGraph sparsification [2] is a widely used strategy to reduce the computational complexity of TSP instances by restricting the search space to a subset of candidate edges. Traditional approaches are mainly based on geometric heuristics, which construct sparse graphs without exploiting instancespecific information.  \nOne of the most commonly used sparsification techniques is the k-nearest neighbor (KNN) graph [3], where each node is connected to its k closest neighbors in terms of Euclidean distance. Another widely adopted sparsification method is Delaunay triangulation [4], which constructs a planar graph by maximizing the minimum angle of all triangles. Delaunay graphs possess strong geometric properties and are known to contain many edges of optimal Euclidean TSP tours in practice. However, although it provides a more structured sparsification compared to KNN graphs, it still includes redundant edges or miss problem-specific structures in certain distributions.  \nIn recent years, learning-based end-to-end approaches have been proposed for TSP [5, 6, 7, 8, 9] . These methods directly learn to construct tours from data, typically using sequence models or attention-based architectures. However, they often exhibit limited generalization capability and involve complex model architectures with a large number of parameters.  \nPreprint.  \nRecent studies have explored learning-based approaches for graph sparsification. Instead of directly constructing tours, these methods aim to identify a subset of promising edges that are likely to appear in high-quality solutions. In particular, Graph Neural Networks (GNNs) can leverage node and edge features to predict edge importance, enabling the construction of sparse graphs that significantly reduce c","cbCaiuYiZ5Fj1TJK","https://ap.wps.com/l/cbCaiuYiZ5Fj1TJK","pdf",2577680,2,1,13,"English","en",105,"# Introduction\n## Graph Sparsification Motivation\n## Prior Sparsification Techniques\n## Learning-based Approaches","[{\"question\":\"Why is exact solving of large-scale Euclidean TSP expensive?\",\"answer\":\"Exact TSP methods require substantial computation time for large instances. Since Euclidean TSP is typically modeled on complete graphs, the number of edges grows quadratically with the number of nodes, creating major overhead for solvers.\"},{\"question\":\"What is Graph Edge Sparsification (GES) in this paper?\",\"answer\":\"GES is a learning-based sparsification approach for Euclidean TSP. It adaptively constructs a sparsification graph for each instance by combining geometric structural information with combinatorial optimization.\"},{\"question\":\"How effective is GES compared with baseline pruning?\",\"answer\":\"Experiments show that GES can prune up to 95% of edges on the MATILDA dataset while keeping the solution gap within 1% of the optimal value. On TSPLIB, pruning can exceed 99% with an optimality gap still below 1%.\"}]",1784201240,33,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"ges-tsp-graph-edge-sparsification-for-tsp","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/ges-tsp-graph-edge-sparsification-for-tsp/85124/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why is exact solving of large-scale Euclidean TSP expensive?","Question",{"text":75,"@type":76},"Exact TSP methods require substantial computation time for large instances. Since Euclidean TSP is typically modeled on complete graphs, the number of edges grows quadratically with the number of nodes, creating major overhead for solvers.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is Graph Edge Sparsification (GES) in this paper?",{"text":80,"@type":76},"GES is a learning-based sparsification approach for Euclidean TSP. It adaptively constructs a sparsification graph for each instance by combining geometric structural information with combinatorial optimization.",{"name":82,"@type":73,"acceptedAnswer":83},"How effective is GES compared with baseline pruning?",{"text":84,"@type":76},"Experiments show that GES can prune up to 95% of edges on the MATILDA dataset while keeping the solution gap within 1% of the optimal value. 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