[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86197-en":3,"doc-seo-86197-105":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},86197,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Approximation Algorithms for Discounted Graph Search with Norm Objectives","The work presents a unified framework for classical graph search and routing problems, including pathwise search, expanding search, minimum spanning tree–like behavior, and traveling salesperson–like behavior. It introduces an edge discount factor α∈[0,1] that reduces costs of repeated edge traversals, and a norm parameter p≥1 that aggregates α-latencies of vertices via the p-norm. The model interpolates between known objectives across endpoints, providing constant-factor approximation guarantees for p=1 and extending them to general p using randomized and derandomized algorithms.","arXiv :2607 . 1 130 1v 1 [ cs .DS] 13 Jul 2026  \nApproximation Algorithms for Discounted Graph Search  \nwith Norm Objectives ∗  \nSvenja M. Griesbach  \nRWTH Aachen University, Department of Computer Science, Germany [griesbach@algo. rwth-aachen. de](griesbach@algo. rwth-aachen. de)  \nFelix Hommelsheim  \nUniversity of Cologne, Department of Computer Science, Germany [hommelsheim@cs. uni-koeln. de](hommelsheim@cs. uni-koeln. de)  \nMax Klimm  \nTechnische Universit¨at Berlin, Institute for Mathematics, Germany [klimm@math. tu-berlin. de](klimm@math. tu-berlin. de)  \nAbstract  \nWe introduce a unified framework for classical search and routing problems, including pathwise search, expanding search, the minimum spanning tree problem, and the traveling salesperson problem. The framework is based on two parameters. The first is a discount factor α ∈ [0 , 1]:  \nthe first traversal of an edge incurs its full cost, whereas each subsequent traversal incurs only an α-fraction of this cost. For a path starting at a designated root vertex, the α-latency of a vertex is the discounted cost accumulated until the vertex is first visited. The second parameter is a norm parameter p ≥ 1. The objective is to find a root-starting path that visits all vertices and minimizes the p-norm of the resulting vector of α-latencies.  \nThe model interpolates between several well-studied objectives. For p = 1 and α = 1, it recovers pathwise search; for p = 1 and α = 0, it recovers expanding search. As p tends to infinity, the objective converges to a makespan-type criterion. At the endpoints α = 1 and α = 0, this limiting objective corresponds to TSP-type and MST-type behavior, respectively. For p = 1, we give polynomial-time constant-factor approximation algorithms for all α ∈ [0 , 1], matching the best known guarantees for expanding search at α = 0 and pathwise search at α = 1 . For general p ≥ 1, we obtain a randomized constant-factor approximation algorithm anda derandomized pseudo-polynomial-time algorithm with the same guarantee.  \n1 Introduction  \nPathwise search, expanding search, the traveling salesperson problem, and the minimum spanning tree problem are four fundamental models for exploring or connecting a network. At first glance, these problems optimize rather different objectives: pathwise and expanding search minimize sums  \n∗ The work of the first author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) by the grant Ho 3831/9-1 (project ID: 514505843) . The work of the second author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) by the grant HO 7562/2-1 (project ID: 573939419) . The work of the third author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany ´s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, EXC-2046/2, project ID: 390685689) . We thank Kevin Schewior for fruitful discussions.  \nof discovery times, while the traveling salesperson problem and the minimum spanning tree problem minimize the time until all vertices are reached or connected. In this paper, we study a common framework that unifies these problems through two parameters. The first parameter determines how repeated traversals of edges are charged, and the second parameter determines how the individual vertex latencies are aggregated.  \nWe are given an undirected graph G = (V, E) with non-negative edge cost ce ∈ N, and a designated start vertex s. A solution is a traversal of the graph, i.e., a path starting in s that may visit edges more than once and eventually visits all vertices. For a vertex v, its latency is the time at which v is visited for the first time. Classical search problems ask for a traversal that minimizes the sum of these latencies.  \nThe pathwise search problem asks for such a traversal when every traversal of an edge e requires ce time units. Thus, the latency of a vertex is equal to the total cost of the prefix","cbCaibYtCSWuC2bN","https://ap.wps.com/l/cbCaibYtCSWuC2bN","pdf",1855093,1,48,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What are the two key parameters in the discounted graph search framework?\",\"answer\":\"The framework uses a discount factor α∈[0,1] to scale the cost of repeated edge traversals and a norm parameter p≥1 to aggregate vertex α-latencies via a p-norm objective.\"},{\"question\":\"How does the model relate to existing search objectives at special parameter values?\",\"answer\":\"For p=1 and α=1 it recovers pathwise search, and for p=1 and α=0 it recovers expanding search. As p→∞, the objective approaches a makespan-type criterion, with behavior corresponding to TSP-type and MST-type endpoints at α=1 and α=0.\"},{\"question\":\"What approximation results are obtained by the authors?\",\"answer\":\"For p=1, they provide polynomial-time constant-factor approximation algorithms for all α∈[0,1]. For general p≥1, they obtain a randomized constant-factor approximation algorithm and a derandomized pseudo-polynomial-time algorithm with the same guarantee.\"}]",1784209336,121,{"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-discounted-graph-search-with-norm-objectives","",{"@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-discounted-graph-search-with-norm-objectives/86197/",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 are the two key parameters in the discounted graph search framework?","Question",{"text":75,"@type":76},"The framework uses a discount factor α∈[0,1] to scale the cost of repeated edge traversals and a norm parameter p≥1 to aggregate vertex α-latencies via a p-norm objective.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the model relate to existing search objectives at special parameter values?",{"text":80,"@type":76},"For p=1 and α=1 it recovers pathwise search, and for p=1 and α=0 it recovers expanding search. As p→∞, the objective approaches a makespan-type criterion, with behavior corresponding to TSP-type and MST-type endpoints at α=1 and α=0.",{"name":82,"@type":73,"acceptedAnswer":83},"What approximation results are obtained by the authors?",{"text":84,"@type":76},"For p=1, they provide polynomial-time constant-factor approximation algorithms for all α∈[0,1]. For general p≥1, they obtain a randomized constant-factor approximation algorithm and a derandomized pseudo-polynomial-time algorithm with the same guarantee.","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"]