[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81675-en":3,"doc-seo-81675-105":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},81675,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Reducing Prize-Collecting Stroll and Related Routing Problems to Prize-Collecting TSP","Prize-collecting stroll is framed as a path version of prize-collecting TSP on complete metric graphs with two fixed terminals and nonnegative vertex penalties for unvisited vertices. A generalization called prize-collecting-Φ-TSP is introduced via prescribed vertices with parity and connectivity requirements. An approximation-preserving reduction is proved: if a δ-approximation for prize-collecting TSP exists, then for any fixed ε>0 a polynomial-time (δ+ε)-approximation for prize-collecting-Φ-TSP follows when prescribed vertices are constant-bounded, yielding improved guarantees for the stroll.","arXiv :2606 . 18 157v 3 [ cs .DS] 10 Jul 2026  \nReducing Prize-Collecting Stroll and Related Routing Problems  \nto Prize-Collecting TSP  \nHong Li  \nSchool of Mathematics and Statistics, Yunnan University  \n[honglimath@126.com](honglimath@126.com)  \nAbstract  \nThe prize-collecting stroll is the path version of the prize-collecting TSP. Given a complete metric graph, two distinct prescribed terminal vertices 􀁂, 􀁃, and nonnegative penalties on vertices, the prize-collecting stroll asks for an 􀁂-􀁃 tour minimizing its length plus the total penalty of vertices that are not visited by it. We study a common generalization of the prize-collecting stroll and several related prize-collecting routing problems, which we call the prize-collecting-Φ -TSP. In this model, Φ specifies a set of prescribed vertices together with their parity and connectivity requirements. We show that, if a 􀁤-approximation algorithm for the prize-collecting TSP is available, then, for every fixed 􀁙 > 0, thereis a polynomial-time (􀁤 + 􀁙)-approximation algorithm for the prize-collecting-Φ-TSP when the number of prescribed vertices is bounded by a fixed constant. Consequently, the prize-collecting stroll can be approximated as well as the prize-collecting TSP up to an arbitrarily small additive loss in the approximation ratio. This yields a better-than-1.6-approximation algorithm for the prize-collecting stroll, improving the previous best-known approximation guarantee of 1.6662.  \n1 Introduction  \n1.1 The PC-Φ-TSP and related prize-collecting routing problems  \nWe first describe the prize-collecting-Φ-TSP (PC-Φ-TSP) . A PC-Φ-TSP instance consists of a complete metric graph 􀀜 = (􀀫, 􀀚) with edge lengths ℓ : 􀀚 → R ≥0, nonnegative vertex penalties 􀁣 : 􀀫 → R ≥0, and an interface Φ = (􀀞, 􀀦, P) . Here 􀀞 ⊆ 􀀫 is a set of prescribed vertices, 􀀦 ⊆ 􀀞 is a set of even cardinality whose vertices are prescribed to have odd degree, and P is a partition of 􀀞 that specifies the required connectivity among the prescribed vertices. The problem asks for a multiset 􀀛 of edges of 􀀜 . Let 􀀫(􀀛) be the set of vertices incident to 􀀛, and regard the vertices in 􀀫 (􀀛) ∪ 􀀞 as visited by 􀀛 . The multigraph (􀀫(􀀛) ∪ 􀀞, 􀀛) is required to satisfy: (i) parity constraint: the vertices in 􀀦 have odd degree and all vertices in (􀀫(􀀛) ∪ 􀀞) \\ 􀀦 have even degree; (ii) 􀀞-connectivity constraint: every connected component contains at least one vertex of 􀀞; and (iii) P-connectivity constraint: for every 􀀘 ∈ P, all vertices of 􀀘 lie in the same connected component. The objective is to minimizeÍ􀀴∈􀀛ℓ(􀀴) +Í􀁅 ∈􀀫\\(􀀫(􀀛 )∪􀀞 ) 􀁣 (􀁅) . Figure 1 illustrates the PC-Φ-TSP.  \nIn this paper, we describe tours as edge multisets rather than as walks. The PC-Φ-TSP generalizes several prize-collecting routing problems. In a complete metric graph 􀀜 = (􀀫, 􀀚) with nonnegative penalties on the vertices in 􀀫, given a root 􀁁 ∈ 􀀫, the prize-collecting TSP (PCTSP) asks for a possibly empty tour rooted at 􀁁 that minimizes its length plus the total penalty of vertices not visited by it. This is exactly the PC-Φ-TSP with 􀀞 = {􀁁} , 􀀦 = ∅, and P = {{􀁁}} . Given two distinct terminals 􀁂, 􀁃 ∈ 􀀫, the prize-collecting stroll (PCS) asks for an 􀁂-􀁃 tour that minimizes its length plus the total penalty of vertices not visited by it. This is exactly the PC-Φ-TSP with 􀀞 = 􀀦 = {􀁂, 􀁃} and P = {{􀁂, 􀁃}} . The PC-Φ-TSP also generalizes the prize-collecting connected 􀀩-join problem [8] and a prize-collecting analogue of the multiple TSP [23] .  \n1.2 Related work  \nThe PCTSP is a special case of a more general problem introduced by Balas [4] . Bienstock, Goemans, Simchi-Levi, and Williamson [5] gave a 2.5-approximation based on threshold rounding, and Goemans and Williamson [12] obtained a 2-approximation by a primal-dual method. After subsequent improvements [2, 6, 11], Blauth, Klein, and Nägele [7] gave the best-known approximation ratio, slightly below 1.6.  \nThe PCS is the path version of the PCTSP and is also known as the prize-collecting path problem.","cbCailwu2nxQ4ekz","https://ap.wps.com/l/cbCailwu2nxQ4ekz","pdf",665385,3,1,16,"English","en",105,"# Abstract\n# Introduction\n## The PC-Φ-TSP and related prize-collecting routing problems\n## Related work","[{\"question\":\"What is the prize-collecting stroll problem being studied?\",\"answer\":\"It seeks a tour between two distinct prescribed terminals that minimizes travel length plus the total penalty of vertices that the tour does not visit.\"},{\"question\":\"How does the paper generalize prize-collecting stroll using the PC-Φ-TSP model?\",\"answer\":\"It defines a set of prescribed vertices together with parity and connectivity constraints specified by an interface Φ, then minimizes the same style of objective over edge multisets satisfying those constraints.\"},{\"question\":\"What approximation relationship does the paper prove between prize-collecting TSP and prize-collecting-Φ-TSP?\",\"answer\":\"If a δ-approximation algorithm for prize-collecting TSP is available, then for any fixed ε\\u003e0 the paper gives a polynomial-time (δ+ε)-approximation for prize-collecting-Φ-TSP when the number of prescribed vertices is bounded by a constant.\"}]",1784175352,40,{"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},"reducing-prize-collecting-stroll-and-related-routing-problems-to-prize-collecting-tsp","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/reducing-prize-collecting-stroll-and-related-routing-problems-to-prize-collecting-tsp/81675/",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},"What is the prize-collecting stroll problem being studied?","Question",{"text":75,"@type":76},"It seeks a tour between two distinct prescribed terminals that minimizes travel length plus the total penalty of vertices that the tour does not visit.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper generalize prize-collecting stroll using the PC-Φ-TSP model?",{"text":80,"@type":76},"It defines a set of prescribed vertices together with parity and connectivity constraints specified by an interface Φ, then minimizes the same style of objective over edge multisets satisfying those constraints.",{"name":82,"@type":73,"acceptedAnswer":83},"What approximation relationship does the paper prove between prize-collecting TSP and prize-collecting-Φ-TSP?",{"text":84,"@type":76},"If a δ-approximation algorithm for prize-collecting TSP is available, then for any fixed ε>0 the paper gives a polynomial-time (δ+ε)-approximation for prize-collecting-Φ-TSP when the number of prescribed 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