[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82004-en":3,"doc-seo-82004-105":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},82004,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Pure Nash Equilibria in Graphical Games of Bounded Width Revisited","Revisiting the parameterized complexity of deciding whether a graphical game admits a pure Nash equilibrium (PNE), the work studies how running time depends on graph parameters such as treewidth and related width measures. The paper first shows that a previously claimed improvement to an αO(tw) dependence is flawed, and that any such algorithm would entail the unlikely collapse FPT=W[1]. It then establishes tighter bounds for pathwidth and cutwidth, using a key relationship between the widths of G, its maximum degree, and the square graph G2.","arXiv :2607 .07627v 1 [ cs .DS] 8 Jul 2026  \nPure Nash Equilibria in Graphical Games of Bounded Width Revisited  \nMichael Lampis \\#   \nLAMSADE, CNRS UMR7243, Université Paris Dauphine-PSL, 75775 Paris, France Yiren Lu \\#   \nLAMSADE, CNRS UMR7243, Université Paris Dauphine-PSL, 75775 Paris, France  \n~~ Abstract ~~  \nWe revisit the complexity of deciding whether a graphical game admits a pure Nash equilibrium (PNE) parameterized by standard measures of the input graph, such as treewidth. The natural dynamic programming algorithm for this problem has parameter dependence α (∆+1)tw where α is the maximum number of strategies available to each player, each player’s utility depends on at most ∆ other players, and the input graph has width tw . Our first contribution is to point out that an algorithm by Thomas and van Leeuwen [Algorithmica 2015] claiming to improve this dependence to αO(tw) is flawed and, more strongly, such an algorithm would imply that FPT=W[1] .  \nWe then set out to pinpoint the fine-grained complexity of this problem with respect to standard parameters and show that the natural DP algorithm is not optimal, as the problem can be solved with dependence α ⌊ 2+1⌋tw , α ⌊ ∆2+1⌋pw, and αctw , where pw, ctw are the pathwidth and cutwidth of the input respectively. Our main algorithmic tool is a tightening of the relationship between the width of a graph G, its maximum degree, and the width of G2 , which may be of independent interest. Complementing these results, we show that our algorithms for pathwidth and cutwidth are likely to be optimal, as improving them is equivalent to falsifying the pw-SETH.  \n2012 ACM Subject Classification Mathematics of computing → Graph algorithms; Theory of Computation → Design and Analysis of Algorithms → Parameterized Complexity and Exact Algorithms; Theory of computation → Algorithmic game theory  \nKeywords and phrases Graphical games, Pure Nash equilibria, Pathwidth, Treewidth  \n 1  Introduction  \nThe computation of Nash equilibria is a central topic at the intersection of computer science and economics. In this paper we focus on equilibria in graphical games, that is, games involving n players represented by the vertices of a (di-)graph whose edges indicate player interactions, in the sense that the presence of an arc (u, v) indicates that the utility of player u (partially) depends on the strategy of player v (and the absence of an arc indicates that u is indifferent to v’s actions) . This is an extremely natural and well-studied model [20, 22 , 23] .  \nThe question we are interested in is the complexity of deciding whether a given graphical game admits a pure Nash equilibrium (PNE), that is, a joint strategy where no player can unilaterally increase her utility by changing her strategy. This problem is NP-complete [33], so we attack it using the tools of parameterized complexity, which is one of the most established approaches for dealing with NP-hardness 1 . Our goal is to investigate the structural parameterized complexity of deciding if a graphical game admits a PNE, for standard parameters such as treewidth, pathwidth, cutwidth and maximum degree.  \nThis question is of course anything but new and in fact its study goes back more than twenty years. In particular, the works of Gottlob, Greco, and Scarcello [12] and Daskalakisand Papadimitriou [8] established the following: if we are given a graphical game G where each player has at most α available strategies, each player’s utility depends on at most ∆  \n1 We assume the reader is familiar with the basics of FPT algorithms as given for example in [6] .  \n2 Pure Nash Equilibria in Graphical Games of Bounded Width Revisited  \nother players, and we are supplied a tree decomposition of the game graph with width tw, then a PNE (if one exists) can be found in time α(∆+1)tw |G| O(1) . Even though the two works arrive at this complexity through different paths (CSPs of bounded hypertreewidth for [12] and Markov Random Fields for [8]), the ","cbCain1odQUYyrqh","https://ap.wps.com/l/cbCain1odQUYyrqh","pdf",748622,7,1,24,"English","en",105,"# Introduction\n## Problem background and parameterized complexity\n## Prior work and claimed improvement\n## Paper goals and main contributions","[{\"question\":\"What is the decision problem studied in the paper?\",\"answer\":\"The paper focuses on deciding whether a given graphical game has a pure Nash equilibrium (PNE), meaning no player can improve her utility by changing her strategy unilaterally.\"},{\"question\":\"Why does the paper consider a previous αO(tw) improvement claim to be flawed?\",\"answer\":\"It argues that the claimed improvement cannot be correct under standard hypotheses: even with α=2, the problem is W[1]-hard parameterized by treewidth (and even vertex cover), which would imply FPT=W[1] if such an algorithm existed.\"},{\"question\":\"Which graph-width parameters does the paper refine for better parameter dependence?\",\"answer\":\"Beyond treewidth, it derives dependence on pathwidth and cutwidth, with running time expressed using tightened exponents tied to ⌊(2+1)tw⌋ and ⌊(Δ2+1)pw⌋-type forms, supported by results relating widths of G, maximum degree, and the square graph 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is the decision problem studied in the paper?","Question",{"text":76,"@type":77},"The paper focuses on deciding whether a given graphical game has a pure Nash equilibrium (PNE), meaning no player can improve her utility by changing her strategy unilaterally.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why does the paper consider a previous αO(tw) improvement claim to be flawed?",{"text":81,"@type":77},"It argues that the claimed improvement cannot be correct under standard hypotheses: even with α=2, the problem is W[1]-hard parameterized by treewidth (and even vertex cover), which would imply FPT=W[1] if such an algorithm existed.",{"name":83,"@type":74,"acceptedAnswer":84},"Which graph-width parameters does the paper refine for better parameter dependence?",{"text":85,"@type":77},"Beyond treewidth, it derives dependence on pathwidth and cutwidth, with running time expressed using tightened exponents tied to ⌊(2+1)tw⌋ and ⌊(Δ2+1)pw⌋-type forms, supported by results relating 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