[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84792-en":3,"doc-seo-84792-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84792,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Approximation Algorithms for the Traveling Thief Problem","The Traveling Thief Problem (TTP) combines the Traveling Salesperson Problem with the Knapsack Problem on a finite metric space with items at locations. The agent must choose a subset of items to collect and a cyclic tour visiting those locations, where collecting items increases profit but slows travel as carried weight grows, yielding two competing objectives: maximize total collected profit and minimize tour travel time. No approximation algorithms were known for any TTP variant, motivating computation of an (α1, α2)-approximate Pareto set. The paper provides a polynomial-time (9+ε, 9+ε)-approximation for the Pareto set and studies a related Weighted TSP case with a (2e+ε)-approximation.","arXiv :2607 .05 164v 1 [ cs .DS] 6 Jul 2026  \nApproximation Algorithms for the Traveling Thief Problem  \nJan Eube \\#   \nUniversity of Bonn, Germany Kelin Luo \\#  University at Buffalo, NY, USA  \nHeiko Röglin \\#   \nUniversity of Bonn, Germany Sarah Sturm \\#  University of Bonn, Germany  \n~~ Abstract ~~  \nThe Traveling Thief Problem (TTP) combines the Traveling Salesperson Problem with the Knapsack Problem. In this problem, a finite metric space is given, and at each location an item with some profit and weight is placed. An agent seeks to collect a subset of the items. To do so, the agent must decide which items to collect and to determine a cyclic tour visiting the corresponding locations. While collecting an item yields its profit as a reward, the agent’s speed decreases as more weight is picked up. The problem involves two competing objectives: maximizing the total profit of the collected items and minimizing the travel time of the tour.  \nWhile many heuristics and exact algorithms (with a non-polynomial running time) have been developed, no approximation algorithms are known for any variant of the TTP. We aim at computing an (α1 , α2 )-approximate Pareto set that, for every solution, contains another solution collecting at least a 1α1 fraction of its profit while requiring at most α2 times its travel time. Our main result isan algorithm that calculates a (9 + ϵ, 9 + ϵ)-approximate Pareto set in polynomial time.  \nWe also consider the setting in which the set of items to be collected is given in advance, so that the agent only has to compute a tour through the corresponding locations that minimizes the total travel time. This is the so-called Weighted TSP. For this setting, we present a (2e +ϵ)-approximation algorithm.  \n2012 ACM Subject Classification Theory of computation → Routing and network design problems  \nKeywords and phrases Traveling Thief Problem, Traveling Salesperson Problem, Knapsack Problem, Approximation Algorithms, Bi-objective optimization  \nFunding This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– 390685813; 459420781 and by the Lamarr Institute for Machine Learning [and Artificial Intelligence lamarr-institute.org](and Artificial Intelligence lamarr-institute.org).  \n 1  Introduction  \nThe Knapsack Problem (KP) and the Traveling Salesperson Problem (TSP) are well-studied combinatorial optimization problems that have applications in many different fields. Both problems are known to be NP-hard. In order to deal with them, one either needs to accept anon-polynomial worst-case running time or rely on heuristics or approximation algorithms. While both problems are already challenging on their own, many real-world applications involve the combination of multiple NP-hard optimization problems. These combinations are often referred to as multi-component optimization problems. In particular, there is a sequence of papers on a combination of the KP and the TSP called the Traveling Thief Problem (TTP) (e.g., [5, 16 , 17 , 13 , 15]) . Following its introduction, there were also multiple  \n2 Approximation Algorithms for the Traveling Thief Problem  \ncompetitions that aimed at solving variants of this problem 1 .  \nAs in the TSP, also in the TTP one is given a set of locations that have to be visited by an agent (the thief) in a cyclic tour. Additionally, there are items placed at the locationsand the agent has to decide which items to pick up. Each item has a profit and a weight and the agent gets slower when it picks up additional weight. The inverse speed is modeled as an increasing function, and the time needed to traverse an edge is obtained by multiplying the current function value by the edge length. There also exists a weight threshold limiting the total weight of the items that the agent is allowed to pick up. Most models in the literature consider the special case in which the agent starts with an initial speed vmax that decreases proportionally to the weight ","cbCaicxZ6vYXrmxy","https://ap.wps.com/l/cbCaicxZ6vYXrmxy","pdf",842230,1,23,"English","en",105,"# Introduction\n# Approximation Algorithms for the Traveling Thief Problem\n## TTP formulation and objectives\n## Approximate Pareto set results\n## Weighted TSP variant","[{\"question\":\"What is the Traveling Thief Problem and how do the two objectives conflict?\",\"answer\":\"TTP merges routing and item selection: the agent chooses items (profit, weight) and a cyclic tour. Collecting more weight slows the agent, so maximizing profit trades off against minimizing travel time.\"},{\"question\":\"What approximation goal does the paper pursue for the TTP?\",\"answer\":\"The paper computes an (α1, α2)-approximate Pareto set, ensuring that for every solution there exists another solution with at least a 1/α1 fraction of its profit while requiring at most α2 times its travel time.\"},{\"question\":\"What approximation guarantees are provided by the main results?\",\"answer\":\"A polynomial-time algorithm computes a (9+ε, 9+ε)-approximate Pareto set for the TTP. For the related setting where the item set is fixed, the paper gives a (2e+ε)-approximation algorithm for minimizing total travel time.\"}]",1784198272,58,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"approximation-algorithms-for-the-traveling-thief-problem","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/approximation-algorithms-for-the-traveling-thief-problem/84792/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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},"What is the Traveling Thief Problem and how do the two objectives conflict?","Question",{"text":75,"@type":76},"TTP merges routing and item selection: the agent chooses items (profit, weight) and a cyclic tour. Collecting more weight slows the agent, so maximizing profit trades off against minimizing travel time.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What approximation goal does the paper pursue for the TTP?",{"text":80,"@type":76},"The paper computes an (α1, α2)-approximate Pareto set, ensuring that for every solution there exists another solution with at least a 1/α1 fraction of its profit while requiring at most α2 times its travel time.",{"name":82,"@type":73,"acceptedAnswer":83},"What approximation guarantees are provided by the main results?",{"text":84,"@type":76},"A polynomial-time algorithm computes a (9+ε, 9+ε)-approximate Pareto set for the TTP. For the related setting where the item set is fixed, the paper gives a (2e+ε)-approximation algorithm for minimizing total travel time.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]