[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84247-en":3,"doc-seo-84247-105":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},84247,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Induced Packing Treewidth","This paper develops a unified framework for graph classes defined by forbidden induced subgraphs or induced minors, linked to classes defined through structured tree decompositions. For a fixed graph family H, it defines induced-H-packing treewidth, where each bag bounds the maximum number of pairwise anticomplete induced copies of graphs from H that intersect the bag. The work derives new quasipolynomial-time algorithmic consequences: MWIS, List 3-Coloring, and Odd Cycle Transversal admit tractable results under specific choices of H, including QPTAS statements.","arXiv :2607 .07595v1 [math .CO] 8 Jul 2026  \nINDUCED PACKING TREEWIDTH  \nAMIR NIKABADI† AND PAWEŁ RZĄŻEWSKI ∥  \nAbstract. In this paper, we introduce a framework that aims to unify classes defined by forbidden induced subgraphs or induced minors with classes defined by the existence of certain structured tree decompositions. Let H be a fixed family of graphs. We define induced-H-packing treewidth, a tree-decomposition-based graph parameter that, for each bag, measures the maximum number of pairwise anticomplete induced copies of graphs from H intersecting that bag. This notion generalizes some previously studied parameters: when H = {P1 }, it is equivalent to tree-independence number, and when H = {P2 }, it is equivalent to induced matching treewidth.  \nWe show that bounded induced-H-packing treewidth yields new algorithmic consequences for a range of choices of H. In particular, we prove the following results for graphs of bounded induced-H-packing treewidth.  \n• If H contains P4 , then Maximum-Weight Independent Set (MWIS) can be solved in quasipolynomial time.  \n• If H contains P5 , then List 3-Coloring can be solved in quasipolynomial time.  \n• If H contains P3 , then Odd Cycle Transversal and several other problems can be solved in quasipolynomial time. Furthermore, there is a quasipolynomialtime approximation scheme (QPTAS) for finding a largest induced subgraph of bounded treewidth that satisfies a given hereditary CMSO2 formula.  \n• If H contains all cycles or any path, then MWIS admits a QPTAS.  \nThese results partially answer and substantially extend a question of Bodlaender, Fomin, and Korhonen [SODA 2026] on the tractability of MWIS for graphs of bounded induced-H-packing treewidth for H = {P3 } and for H equal to the family of all cycles.  \n†IT University of Copenhagen, Denmark ([amir@itu.dk](amir@itu.dk)) . Supported by the Independent Research Fund Denmark (DFF), grant agreement number 2098-00012B.  \n∥ Warsaw University of Technology, Poland ([pawel.rzazewski@pw.edu.pl](pawel.rzazewski@pw.edu.pl)) . Supported by the National Science Centre grant 2024/54/E/ST6/00094 .  \nContents  \n1. Introduction 1  \n1.1. Our contribution 3  \n2. Technical overview 5  \n3. Preliminaries 10  \n4. Container lemmas 13  \n5. MWIS in graphs of bounded induced-P4-packing treewidth 16  \n5.1. Dealing with (B, P4 )-clean instances 16  \n5.2. Wrapping up 18  \n6. List 3-Coloring in graphs of bounded induced-P5-packing treewidth 19  \n6.1. Dealing with (B, P5 )-clean instances. 19  \n6.2. Wrapping up 23  \n7. Algorithms for graphs of bounded induced-P3-packing treewidth 24  \n7.1. Odd Cycle Transversal 24  \n7.2. Blob graphs and their applications 27  \n8. Induced packing treewidth and dominated balanced separators 29  \n8.1. Induced packing of long paths 30  \n8.2. Induced packing of cycles 30  \n9. Computing decompositions 32  \n9.1. Approximation algorithm 32  \n9.2. Hardness of computing induced packing treewidth 34  \n10. Conclusion 35  \nReferences 36  \nINDUCED PACKING TREEWIDTH 1  \n1. Introduction  \nAmong classic computational graph problems, Max Independent Set (and its weighted variant) and 3-Coloring (and its list variant) are arguably two of the best studied ones. In the Max Weight Independent Set (MWIS) problem, we are given a graph G and a weight function w : V (G) → Q≥0, and the goal is to find an independent set (i.e., a set of pairwise non-adjacent vertices) of maximum total weight. In the List 3-Coloring problem, we are given a graph G and a list assignment L : V (G) → 2 {1 ,2 ,3} , and the goal is to find a proper coloring of G such that each vertex v is colored with a color from its list L(v) . Both problems are among Karp’s 21 NP-hard [42] problems and remain hard for many notions of hardness considered in complexity theory [13 , 19 , 26 , 36 , 41 , 43 , 47] . This persistent hardness naturally leads to a central question in algorithmic graph theory: Which restrictions on the input instances make MWIS and List 3-Coloring tractable?  \nForbi","cbCaisHcJs8Mbm4g","https://ap.wps.com/l/cbCaisHcJs8Mbm4g","pdf",694254,5,1,41,"English","en",105,"# Introduction\n## Our contribution\n# Technical overview\n# Preliminaries\n# Container lemmas\n# MWIS in graphs of bounded induced-P4-packing treewidth\n## Dealing with (B, P4)-clean instances\n## Wrapping up\n# List 3-Coloring in graphs of bounded induced-P5-packing treewidth\n## Dealing with (B, P5)-clean instances\n## Wrapping up\n# Algorithms for graphs of bounded induced-P3-packing treewidth\n## Odd Cycle Transversal\n## Blob graphs and their applications\n# Induced packing treewidth and dominated balanced separators\n## Induced packing of long paths\n## Induced packing of cycles\n# Computing decompositions\n## Approximation algorithm\n## Hardness of computing induced packing treewidth\n# Conclusion","[{\"question\":\"What is induced-H-packing treewidth and how is it defined?\",\"answer\":\"It is a tree-decomposition-based parameter for a fixed family of graphs H. For each bag, it measures the maximum number of pairwise anticomplete induced copies of graphs from H that intersect that bag.\"},{\"question\":\"Which graph problems become solvable in quasipolynomial time under bounded induced-H-packing treewidth?\",\"answer\":\"When H contains P4, MWIS is solvable in quasipolynomial time; when H contains P5, List 3-Coloring is solvable in quasipolynomial time; and when H contains P3, Odd Cycle Transversal and several other problems admit quasipolynomial-time solutions.\"},{\"question\":\"Does the paper provide approximation guarantees?\",\"answer\":\"Yes. It states a quasipolynomial-time approximation scheme (QPTAS) for finding a largest induced subgraph of bounded treewidth satisfying a given hereditary CMSO2 formula, and it also gives a QPTAS for MWIS when H contains all cycles or any path.\"}]",1784194340,103,{"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},"induced-packing-treewidth","",{"@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/induced-packing-treewidth/84247/",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","2026-07-28","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 is induced-H-packing treewidth and how is it defined?","Question",{"text":76,"@type":77},"It is a tree-decomposition-based parameter for a fixed family of graphs H. For each bag, it measures the maximum number of pairwise anticomplete induced copies of graphs from H that intersect that bag.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Which graph problems become solvable in quasipolynomial time under bounded induced-H-packing treewidth?",{"text":81,"@type":77},"When H contains P4, MWIS is solvable in quasipolynomial time; when H contains P5, List 3-Coloring is solvable in quasipolynomial time; and when H contains P3, Odd Cycle Transversal and several other problems admit quasipolynomial-time solutions.",{"name":83,"@type":74,"acceptedAnswer":84},"Does the paper provide approximation guarantees?",{"text":85,"@type":77},"Yes. It states a quasipolynomial-time approximation scheme (QPTAS) for finding a largest induced subgraph of bounded treewidth satisfying a given hereditary CMSO2 formula, and it also gives a QPTAS for MWIS when H contains all cycles or any path.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]