[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85384-en":3,"doc-seo-85384-105":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},85384,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Solving the General Case of Rank-3 Maker-Breaker Games in Polynomial Time","A rank-3 Maker-Breaker game is defined on a hypergraph where every hyperedge contains at most three vertices. Maker withdraws vertices to cover an entire hyperedge, while Breaker deletes all hyperedges containing a chosen vertex; winning depends on whether a hyperedge is fully withdrawn or all hyperedges are removed. The work establishes that the general rank-3 case, previously only known for degenerate ranks, admits a polynomial-time solution. The approach derives optimal play using a minimax algorithm supported by theoretical bounds on shortened-game length.","SOLVING THE GENERAL CASE OF RANK-3 MAKER-BREAKER  \nGAMES IN POLYNOMIAL TIME  \nLEAR BAHACK  \n[lear.bahack@gmail.com](lear.bahack@gmail.com)  \nAbstract. A rank-3 Maker-Breaker game is played on a hypergraph in which all hyperedges are sets of at most 3 vertices. The two players of the game, called Maker and Breaker, move alternately. On his turn, maker chooses a vertex to be withdrawn from all hyperedges, while Breaker on her turn chooses a vertex and delete all the hyperedges containing that vertex. Maker wins when by the end of his turn some hyperedge is completely covered, i.e. the last remaining vertex of that hyperedge is withdrawn. Breaker wins when by the end of her turn, all hyperedges have been deleted.  \nSolving a Maker-Breaker game is the computational problem of choosing an optimal move, or equivalently, deciding which player has a winning strategy in a configuration. The complexity of solving two degenerate cases of rank-3 games has been proven before to be polynomial. In this paper, we show that the general case of rank-3 Maker-Breaker games is also solvable in polynomial time.  \nAlgorithms in Graphs; Combinatorial Games; Computational Complexity; Complexity of Games; Game Theory; Hypergraphs; Maker-Breaker; POS-CNF Problem.  \n1. Introduction  \nThe Maker-Breaker game is a positional game, firstly introduced in the classical paper[4] that proves sufficient (but unnecessary) winning conditions for both Maker and Breaker. While Maker-Breaker games are mostly researched in extremal combinatorics and similar contexts, a few important papers[1, 2, 9, 10] deal with the computational complexity of solving the games.  \nShaefer[10] was the first to show a (polynomial) reduction from solving rank-11 Maker-Breaker games to solving general TQBF problems, thus proving that solving the general Maker-Breaker game is PSPACE-complete. This result is of the highest significance in the field of games complexity, since for many combinatorial games establishing a reduction to the Maker-Breaker game is far more natural than to classical PSPACE-hard computational problems. The Maker-Breaker game, thus, is a major tool in the research of other games complexity. We note that Shaefer and others often use an equivalent formulation of the Maker-Breaker game, in terms of positive CNF formulas, instead of the hypergraphs formulation used here.  \nOur two-parts work closes the existing gap between rank-6 games (that are PSPACEcomplete to solve[9]) and two degenerate cases of rank-3 games that are solved in  \n2 LEAR BAHACK  \npolynomial time[8, 7] . In this paper, we solve the (general case) rank-3 game in polynomial time, and in the complementing paper[1] we prove that solving rank-4 games is a PSPACE-complete problem.  \n1.1. Related Works. Lutfar and Watson[8] solved (in polynomial time) rank-3 games in which every edge of the hypergraph contains a unique vertex that is not shared by other edges. Their technique is strongly dependent on those unique vertices, and cannot be generalized. However, their algorithms is not merely in P, but in the complexity class L.  \nKutz[7] solved (in polynomial time) rank-3 games in which distinct edges may share at most one vertex in common. While this case is also a degenerate case, and dealing with pairs of edges sharing two vertices imposes great challenges, we have taken a similar course of thought. In fact, we believe Kutz have been close to solving the general case. However, his style of branching into numerous cases, makes any adaptation to the general case too difficult (at least for humans) .  \n1.2. Our Technique and Structure of the Paper. In our work, we develop some tools and general theory of rank-3 games. We manage to keep the number of separate cases in our proof relatively small, by systematically applying our theoretical tools to analyze the structure of the games considered. Yet the reader is warned that this paper is rather technical.  \nWe define a shortened version of the Maker-Breaker game, by introducin","cbCaicN5fCG0mcGW","https://ap.wps.com/l/cbCaicN5fCG0mcGW","pdf",392375,3,1,16,"English","en",105,"# Introduction\n## Related Works\n## Our Technique and Structure of the Paper\n# Definitions and Lemmata\n## Hypergraphs","[{\"question\":\"What are the rules for winning in a rank-3 Maker-Breaker game?\",\"answer\":\"Maker wins when a hyperedge becomes completely covered, meaning its last remaining vertex is withdrawn on Maker’s turn. Breaker wins when all hyperedges have been deleted by the end of her turns.\"},{\"question\":\"What computational problem does the paper address?\",\"answer\":\"Solving a Maker-Breaker game means choosing an optimal move, equivalently determining which player has a winning strategy for a given configuration.\"},{\"question\":\"What is the main result regarding rank-3 games?\",\"answer\":\"The paper shows that the general case of rank-3 Maker-Breaker games is solvable in polynomial time, closing the gap between known degenerate polynomial cases and other ranks.\"}]",1784203049,40,{"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},"solving-the-general-case-of-rank-3-maker-breaker-games-in-polynomial-time","",{"@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/solving-the-general-case-of-rank-3-maker-breaker-games-in-polynomial-time/85384/",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 are the rules for winning in a rank-3 Maker-Breaker game?","Question",{"text":75,"@type":76},"Maker wins when a hyperedge becomes completely covered, meaning its last remaining vertex is withdrawn on Maker’s turn. Breaker wins when all hyperedges have been deleted by the end of her turns.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What computational problem does the paper address?",{"text":80,"@type":76},"Solving a Maker-Breaker game means choosing an optimal move, equivalently determining which player has a winning strategy for a given configuration.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main result regarding rank-3 games?",{"text":84,"@type":76},"The paper shows that the general case of rank-3 Maker-Breaker games is solvable in polynomial time, closing the gap between known degenerate polynomial cases and other ranks.","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":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"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":29,"slug":118},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]