[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81749-en":3,"doc-seo-81749-105":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},81749,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Temporal Path Covers Dilworth Properties and Parameterized Complexity","Minimum Temporal Path Cover (TPC) and Minimum Temporally Disjoint Path Cover (TDPC) are studied on temporal DAGs, with earlier results showing NP-hardness while tractable cases satisfy a temporal Dilworth property. The paper proves that, under the corresponding promise, the minimum size equals the Lovász number of the connectivity graph for both problems. It also derives parameterized algorithms and hardness: TPC is W[1]-hard via deletion distance to linear forest even with two time-steps, while FPT is achievable using vertex cover number, and time-steps are necessary for tractability.","Temporal Path Covers: Dilworth Properties and Parameterized Complexity  \nLapo Cioni \\#   \nDepartment of Statistic, Computer Science and Applications, Università degli Studi di Firenze, Firenze, Italy  \nSotiris Kanellopoulos \\#   \nNational Technical University of Athens and Archimedes, Athena Research Center Edouard Nemery \\#   \nUniversité Paris Dauphine – PSL, CNRS UMR7243, LAMSADE, Paris, France Aris Pagourtzis \\#   \nNational Technical University of Athens and Archimedes, Athena Research Center Christos Pergaminelis \\#   \nNational Technical University of Athens and Archimedes, Athena Research Center Manolis Vasilakis \\# Ñ  \nUniversité Paris Dauphine – PSL, CNRS UMR7243, LAMSADE, Paris, France  \n~~ Abstract ~~  \nThe Minimum Temporal Path Cover (TPC) and Minimum Temporally Disjoint Path Cover (TDPC) problems were introduced by [Chakraborty, Dailly, Foucaud, Klasing, MFCS ’24] . Both were shown to be NP-hard on temporal DAGs, while the latter is also NP-hard on temporal oriented trees. All tractable cases for T(D)PC established in that paper satisfy a temporal Dilworth property, namely that the size of the minimum T(D)PC is equal to the size of the maximum antichain. This raises a natural question: is T(D)PC polynomial-time solvable under the promise that the respective Dilworth property holds? In this work, we answer this question in the affirmative for both problems, proving in fact that, under the respective promise, the size of the minimum T(D)PC is exactly equal to the Lovász number of the connectivity graph.  \nIn another direction, we establish parameterized algorithms and hardness results for TPC and TDPC. Our main result is that TPC is W[1]-hard parameterized by the deletion distance to linear forest even for temporal graphs with two time-steps, answering in the negative an open question by Chakraborty et al. about whether an XP algorithm parameterized by treewidth plus number of time-steps can be improved to FPT. On the other hand, we prove that an FPT algorithm does exist if the vertex cover number is used as parameter instead of the treewidth in the above parameterization. We complement this with a proof that including the number of time-steps in the parameter is necessary to yield tractability, as, otherwise, both TPC and TDPC remain NP-hard even for constant vertex cover size. Along the way, we establish various other para-NP-hardness results involving structural parameters such as the pathwidth and the maximum degree of the underlying graph.  \n2012 ACM Subject Classification Theory of computation → Problems, reductions and completeness; Theory of computation → Parameterized complexity and exact algorithms  \nKeywords and phrases Temporal Graphs, Dilworth Property, Lovász Number, Parameterized Complexity  \n2 Temporal Path Covers: Dilworth Properties and Parameterized Complexity  \n 1  Introduction  \nTemporal graphs extend the concept of static graph to include time-dependent information, with transitions through edges being possible only if the respective edge is active in the current time-step. As such, temporal graphs naturally model scenarios such as road networks, bus or train transits, and flight routes. Recent research has focused on pathfinding [7, 36], temporal walks and exploration [2, 17], temporal flows and cuts [1] and reachability [9, 10, 11, 13], among others. For a comprehensive overview on temporal graphs, we refer to [26, 35] .  \nIn an MFCS 2024 paper, Chakraborty, Dailly, Foucaud and Klasing [5] introduced the Minimum Temporal Path Cover (TPC) and Minimum Temporally Disjoint Path Cover (TDPC) problems as the temporal analogues of Minimum Path Cover and Minimum Path Partition respectively (cf. [15, 19, 28]), with both problems being NP-hard even in temporal DAGs. Interestingly, all temporal graph classes with polynomialtime solvable T(D)PC established in that paper are shown to satisfy a (temporal) Dilworth property: the size of the minimum T(D)PC is guaranteed to be equal to the size of the maximum (","cbCait0uySGcdg8Q","https://ap.wps.com/l/cbCait0uySGcdg8Q","pdf",1069384,3,1,24,"English","en",105,"# Introduction\n## Temporal path covers and Dilworth property\n## Main results and parameterized complexity","[{\"question\":\"What are Minimum Temporal Path Cover (TPC) and Minimum Temporally Disjoint Path Cover (TDPC)?\",\"answer\":\"They are temporal analogues of classical path covering and path partitioning problems, defined on temporal graphs where edges are usable only at their active time-steps.\"},{\"question\":\"How does the Dilworth property affect solvability for TPC and TDPC?\",\"answer\":\"Under the promise that the relevant Dilworth property holds, the paper shows the minimum T(D)PC size equals the Lovász number of the connectivity graph, yielding tractability.\"},{\"question\":\"What parameterized complexity results are proved for TPC?\",\"answer\":\"TPC is W[1]-hard when parameterized by deletion distance to linear forest even for temporal graphs with two time-steps, ruling out certain FPT possibilities; however, an FPT algorithm exists when parameterized by vertex cover number, and including the number of time-steps is necessary.\"}]",1784175819,60,{"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},"temporal-path-covers-dilworth-properties-and-parameterized-complexity","",{"@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/temporal-path-covers-dilworth-properties-and-parameterized-complexity/81749/",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","2026-07-26","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 Minimum Temporal Path Cover (TPC) and Minimum Temporally Disjoint Path Cover (TDPC)?","Question",{"text":75,"@type":76},"They are temporal analogues of classical path covering and path partitioning problems, defined on temporal graphs where edges are usable only at their active time-steps.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the Dilworth property affect solvability for TPC and TDPC?",{"text":80,"@type":76},"Under the promise that the relevant Dilworth property holds, the paper shows the minimum T(D)PC size equals the Lovász number of the connectivity graph, yielding tractability.",{"name":82,"@type":73,"acceptedAnswer":83},"What parameterized complexity results are proved for TPC?",{"text":84,"@type":76},"TPC is W[1]-hard when parameterized by deletion distance to linear forest even for temporal graphs with two time-steps, ruling out certain FPT possibilities; 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