[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83745-en":3,"doc-seo-83745-105":30,"detail-sidebar-cat-0-en-105":92},{"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},83745,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","TSP with Predictions: Heatmap to Tour with Provable Guarantees","The Traveling Salesperson Problem (TSP) serves as a benchmark for optimization techniques, and has become a key testbed for machine learning methods that predict solution structure. A prevalent approach predicts a heatmap estimating which edges belong to an optimal tour, but converting the heatmap into an actual tour is difficult and costly. This work presents algorithms that transform heatmaps into tours while proving that the approximation quality depends explicitly on the heatmap’s L1 distance to an optimal solution. Experiments compare the method against prior heatmap-based approaches.","TSP with Predictions: Heatmap to Tour with Provable Guarantees  \narXiv :2607 .0379 1v 1 [ cs .DS] 4 Jul 2026  \nMarek Eliáš  \n[marek.elias@unibocconi.it](marek.elias@unibocconi.it)  \nFabrizio Grandoni [fabrizio@idsia.ch](fabrizio@idsia.ch)  \nEleonora Vercesi  \n[eleonora.vercesi@usi.ch](eleonora.vercesi@usi.ch)  \nAdam Polak  \n[adam.polak@unibocconi.it](adam.polak@unibocconi.it)  \nAbstract  \nThe Traveling Salesperson Problem (TSP) has long served as a benchmark for evaluating the strength of optimization techniques in the classical theory of algorithms. In recent efforts to apply ML to algorithmic problems, TSP has also become a natural testbed for the development of ML-based techniques. A common approach is to train a neural network to output a heatmap estimating the likelihood of each edge to be part of the optimal tour; however, converting such a heatmap into an actual tour remains a non-trivial and often computationally intensive step. In this work, we propose algorithms for transforming heatmaps into tours with theoretical guarantees linking the achieved approximation ratio to the quality of the provided heatmap. In the spirit of algorithms with predictions, our results can be described as (1 + 2η/OPT)-approximation algorithms, where η denotes the L1 distance between the prediction (heatmap) and an optimal solution (tour) . Since the previous works lack such explicit guarantees, we compare our approach against them experimentally.  \n1 Introduction  \nWe study the Traveling Salesperson Problem (TSP), one of the key problems in combinatorial optimization: We are given a road network between n cities represented by an undirected edge-weighted graph G. Our task is to find a closed walk in G of minimum total weight that visits every city (at least once) .  \nOur main result is an algorithm that takes as input a TSP instance and additionally, for each edge, a number between 0 and 1 denoting how likely this edge is to belong to an optimal TSP tour. These additional numbers can be understood as predictions or a heatmap. The algorithm outputs a tour that is guaranteed tobe at most 2η longer than an optimal tour, where η denotes the L1 distance between the predictions vector and a groundtruth optimal tour.  \nOur contribution can be understood in two contexts: a theoretical context of the so-called algorithms with predictions (a.k.a. learning-augmented algorithms) [Mitzenmacher and Vassilvitskii, 2020, Lindermayr and Megow, 2022], and a practical context of the so-called solution search stage in the neural combinatorial optimization pipeline [Joshi et al., 2022] .  \nAlgorithms with predictions. This line of work tries to combine provable worst-case guarantees of classical algorithms with great beyond worst-case performance of ML models. The idea is to have a classical algorithm taking as additional input an uncertain hint –prediction of an ML model – and have its performance (e.g., running time or solution cost) bounded by a function of the error of the prediction. Despite almost 400 papers [Lindermayr and Megow, 2022] developed over the last few years, approximation algorithms for TSP have not been studied in this paradigm. We address this gap in the literature.  \nNeural combinatorial optimization. It is a big question if and how neural networks (NNs) can help solving combinatorial optimization problems with many hard constraints, which are usually hard to impose on NN’s output [Bengio et al., 2021] . TSP has long served as a benchmark for combinatorial optimization techniques, and it has a long history of hand-optimized solvers [Applegate and Cook, 2006, Helsgaun, 2000] . So it is interesting to understand if NNs can beat them. A common neural optimization approach to TSP – formalized by Joshi et al. [2022] into a five-stages pipeline – is to have a (graph) NN that assigns to each edge the probability that it belongs to an optimal solution (the so-called heatmap), followed by a “solution search” algorithm that turns these probabilities","cbCaisWG0t3nemRp","https://ap.wps.com/l/cbCaisWG0t3nemRp","pdf",1357584,5,1,36,"English","en",105,"# Introduction\n## Algorithms with predictions\n## Neural combinatorial optimization\n## Our results","[{\"question\":\"What problem does the paper address in the neural TSP pipeline?\",\"answer\":\"It addresses the challenge of turning a predicted edge heatmap into a feasible TSP tour, a step that is often non-trivial and computationally intensive and that prior methods lack explicit guarantees for.\"},{\"question\":\"How does the proposed guarantee relate to the quality of the heatmap?\",\"answer\":\"The algorithm outputs a tour whose cost is provably close to optimal, with the bound expressed through η, the L1 distance between the predicted heatmap (probabilities over edges) and a ground-truth optimal tour.\"},{\"question\":\"How do the authors position their contribution against existing approaches?\",\"answer\":\"They compare experimentally against prior heatmap-based techniques, and interpret the method both as learning-augmented algorithms with theoretical performance tied to prediction error and as an improved solution-search stage in neural combinatorial optimization.\"}]",1784190174,91,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"tsp-with-predictions-heatmap-to-tour-with-provable-guarantees","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/tsp-with-predictions-heatmap-to-tour-with-provable-guarantees/83745/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does the paper address in the neural TSP pipeline?","Question",{"text":76,"@type":77},"It addresses the challenge of turning a predicted edge heatmap into a feasible TSP tour, a step that is often non-trivial and computationally intensive and that prior methods lack explicit guarantees for.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the proposed guarantee relate to the quality of the heatmap?",{"text":81,"@type":77},"The algorithm outputs a tour whose cost is provably close to optimal, with the bound expressed through η, the L1 distance between the predicted heatmap (probabilities over edges) and a ground-truth optimal tour.",{"name":83,"@type":74,"acceptedAnswer":84},"How do the authors position their contribution against existing approaches?",{"text":85,"@type":77},"They compare experimentally against prior heatmap-based techniques, and interpret the method both as learning-augmented algorithms with theoretical performance tied to prediction error and as 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