[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85821-en":3,"doc-seo-85821-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},85821,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Approximate Colorwise Tensorization of Entropy and Optimal Mixing of the Wang-Swendsen-Kotecký Dynamics","Studies the mixing time of Wang–Swendsen–Kotecký (WSK) dynamics for uniformly sampling proper k-colorings. Develops new analysis tools using relative entropy contraction, including criteria for multi-spin distributions: approximate colorwise tensorization of entropy (ACTE) and approximate colorwise subadditivity of entropy (ACSE). These criteria form a colorwise counterpart to standard vertex-wise entropy factorization and enable local-to-global inductive proofs. Applies the framework to obtain an optimal (log k) mixing time for chordal and outerplanar graphs, covering tree vertex and edge colorings.","arXiv :2607 . 10 1 19v 1 [ cs .DS] 11 Jul 2026  \nAPPROXIMATE COLORWISE TENSORIZATION OF ENTROPY AND OPTIMAL MIXING OF THE WANG–SWENDSEN–KOTECK DYNAMICS  \nCHUNYANG WANG, YUICHI YOSHIDA, ZIHAN ZHANG  \nAbstract. We study the mixing time of Wang–Swendsen–Koteck (WSK) dynamics for uniformly sampling proper 􀁀 -colorings. The WSK dynamics is widely used in statistical physics for sampling from the antiferromagnetic Potts model and can be considered a global counterpart of the flip dynamics, which currently yields the state-of-the-art bounds for sampling colorings in general graphs (Carlson and Vigoda, SODA 2025) . However, despite its importance, the tools for analyzing such dynamics remain limited.  \nWe develop new tools that enable us to analyze the mixing time ofthe WSK dynamics through the lens of relative entropy contraction. We introduce new criteria for multi-spin distributions: approximate colorwise tensorization of entropy (ACTE) and approximate colorwise subadditivity of entropy (ACSE) . These criteria provide a colorwise counterpart to standard vertex-wise entropy factorization principles, and expose a form of color symmetry beyond coordinate-wise analyses. We also develop new inductive approaches for establishing such criteria on specific types of graphs, which can be viewed as local-to-global arguments for proving high-dimensional functional inequalities in a graph-theoretic sense.  \nAs concrete applications, we establish an optimal 􀀤􀁀 (log 􀀽) mixing time for the WSK dynamics on chordal and outerplanar graphs, down to the optimal number of colors. Because trees and line graphs of trees are chordal, the result covers both vertex and edge colorings of trees. Our results work in a regime that bypasses the irreducibility threshold for Glauber dynamics while also improving the best known mixing time bounds (Carlson, Chen, Feng and Vigoda, SODA 2025) .  \nContents  \n1. Introduction 1  \n2. Preliminaries and Notation 7  \n3. Colorwise Localization Scheme for the WSK dynamics 12  \n4. Inductive Proofs Establishing Approximate Colorwise Tensorization of Entropy 13  \n5. A Lower Bound on the Mixing Time ofthe WSK Dynamics 23  \n6. Conclusions and Open Problems 27  \nAcknowledgements 27  \nReferences 27  \n1. Introduction  \nSampling proper 􀁀 -colorings remains one of the most fundamental yet notoriously difficult problems in the field of approximate counting and sampling. A longstanding open conjecture postulates the existence of an efficient algorithm for approximately counting and sampling proper 􀁀 -colorings of graphs with maximum degree Δ whenever 􀁀 ≥ Δ + 1. Despite a storied history, we are still far from proving this conjecture. To date, the best threshold for efficient approximate counting and sampling is 􀁀 > 1. 809Δ, recently established by Carlson and Vigoda [CV25] . This result represents the frontier of a line of work [Vig99, CDM+19, CV25] analyzing a Markov chain known as flip dynamics, wherein a two-color Kempe component is chosen and potentially flipped at each step.  \nAn alternative Markov chain for sampling proper 􀁀 -colorings is the Wang–Swendsen–Koteck´y (WSK) dynamics [WSK89, WSK90], which is also based on Kempe moves and can be viewed as a global counterpart of the flip dynamics. The WSK dynamics originated in statistical physics as a cluster algorithm for the antiferromagnetic 􀁀-state Potts model. Although it is less studied from a theoretical  \n(Chunyang Wang, Yuichi Yoshida, Zihan Zhang) National Institute of Informatics, Tokyo, Japan. Email:{c wang,yyoshida,[zihan](zihan}@nii.ac.jp)[}](zihan}@nii.ac.jp)[@nii.ac.jp](zihan}@nii.ac.jp)  \nperspective, it is widely used in practice and shown to be empirically effective on specific instances. Numerical studies [FS99, Sok01] have suggested that the WSK dynamics might exhibit constant-time mixing for the critical [ABC+21] 􀁀 = 3 regime on 􀀽 × 􀀽 periodic square lattices with even 􀀽 .  \nThe WSK dynamics for sampling proper 􀁀 -colorings is formally defined as follows:  \nDefiniti","cbCaisqISCiV6Ca5","https://ap.wps.com/l/cbCaisqISCiV6Ca5","pdf",589665,1,30,"English","en",105,"# Introduction\n# Preliminaries and Notation\n# Colorwise Localization Scheme for the WSK dynamics\n# Inductive Proofs Establishing Approximate Colorwise Tensorization of Entropy\n# A Lower Bound on the Mixing Time of the WSK Dynamics\n# Conclusions and Open Problems","[{\"question\":\"What problem does the document address regarding WSK dynamics?\",\"answer\":\"It studies the mixing time of the Wang–Swendsen–Kotecký (WSK) dynamics for uniformly sampling proper k-colorings of graphs.\"},{\"question\":\"What new tools or criteria are introduced to analyze mixing time?\",\"answer\":\"It introduces relative-entropy-contraction-based tools, including approximate colorwise tensorization of entropy (ACTE) and approximate colorwise subadditivity of entropy (ACSE).\"},{\"question\":\"Which graph classes receive optimal mixing-time results and what do they cover?\",\"answer\":\"The results establish an optimal (log k) mixing time on chordal and outerplanar graphs, which implies coverage of both vertex and edge colorings of trees since trees and line graphs of trees are chordal.\"}]",1784206455,76,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"approximate-colorwise-tensorization-of-entropy-and-optimal-mixing-of-the-wang-swendsen-kotecky-dynamics","",{"@graph":35,"@context":84},[36,53,67],{"@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/approximate-colorwise-tensorization-of-entropy-and-optimal-mixing-of-the-wang-swendsen-kotecky-dynamics/85821/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the document address regarding WSK dynamics?","Question",{"text":74,"@type":75},"It studies the mixing time of the Wang–Swendsen–Kotecký (WSK) dynamics for uniformly sampling proper k-colorings of graphs.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What new tools or criteria are introduced to analyze mixing time?",{"text":79,"@type":75},"It introduces relative-entropy-contraction-based tools, including approximate colorwise tensorization of entropy (ACTE) and approximate colorwise subadditivity of entropy (ACSE).",{"name":81,"@type":72,"acceptedAnswer":82},"Which graph classes receive optimal mixing-time results and what do they cover?",{"text":83,"@type":75},"The results establish an optimal (log k) mixing time on chordal and outerplanar graphs, which implies coverage of both vertex and edge colorings of trees since trees and line graphs of trees are 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