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On the Complexity of Computing Winning Strategies for Finite Poset Games

Michael Soltys, Craig Wilson
2010 Theory of Computing Systems  
This paper is concerned with the complexity of computing winning strategies for poset games.  ...  While it is reasonably clear that such strategies can be computed in PSPACE, we give a simple proof of this fact by a reduction to the game of geography.  ...  In order to study the complexity of poset games and their reductions, we assume that the posets are finite (i.e., U is finite).  ... 
doi:10.1007/s00224-010-9254-y fatcat:y5gcoqb225crvheofsd7gpicpi

Deciding the Winner of an Arbitrary Finite Poset Game Is PSPACE-Complete [chapter]

Daniel Grier
2013 Lecture Notes in Computer Science  
However, until recently the complexity of arbitrary finite poset games was only known to exist somewhere between NC^1 and PSPACE.  ...  The first player unable to select an element of the poset loses. Polynomial time algorithms exist for certain restricted classes of poset games, such as the game of Nim.  ...  I am also very grateful for the support I received from the University of South Carolina Honors College and for all of those who helped me edit and refine this paper.  ... 
doi:10.1007/978-3-642-39206-1_42 fatcat:e6afio2x6nbdnksg6dk5cfvrwq

Combinatorial Game Complexity: An Introduction with Poset Games [article]

Stephen A. Fenner, John Rogers
2015 arXiv   pre-print
Poset games have been the object of mathematical study for over a century, but little has been written on the computational complexity of determining important properties of these games.  ...  In this introduction we develop the fundamentals of combinatorial game theory and focus for the most part on poset games, of which Nim is perhaps the best-known example.  ...  In discussing game complexity, we will abuse terminology and refer to a family of games simply as a game. (The same abuse occurs in other areas of complexity, notably circuit complexity.)  ... 
arXiv:1505.07416v2 fatcat:yxcmoncmpvailp27irc7e3ydbi

Pomax games - a family of partizan games played on posets [article]

Erik Järleberg, Jonas Sjöstrand
2014 arXiv   pre-print
However, for pomax games on general posets of height 3 we show that the problem of deciding the winner is PSPACE-complete and for posets of height 2 we prove NP-hardness.  ...  We prove that pomax games are always integer-valued and for colored tree posets and chess-colored Young diagram posets we give a simple formula for the value of the game.  ...  However, the similarity with restrained Hackenbush apparently disappears for more complex posets (or more complex Hackenbush graphs).  ... 
arXiv:1405.1914v1 fatcat:jagpx53c6jh25jmaat5vuac64u

Flipping the winner of a poset game

Adam O. Kalinich
2012 Information Processing Letters  
We investigate the complexity of computing which player of a poset game has a winning strategy.  ...  This construction also allows us to reduce the class of Boolean formulas to poset games, establishing a lower bound on the complexity of poset games.  ...  Acknowledgements This work would not have happened without the guidance and supervision of Dr. Lance Fortnow. He introduced me to the problem and advised me in preparing this paper.  ... 
doi:10.1016/j.ipl.2011.09.016 fatcat:gowu7k4u2ze7rk4y2rqdb43ibi

Strategy-Stealing is Non-Constructive [article]

Greg Bodwin, Ofer Grossman
2019 arXiv   pre-print
We prove that this problem is PSPACE-hard already for Minimum Poset Games and Symmetric Maker-Maker Games, which are simple classes of games that capture two of the main types of strategy-stealing arguments  ...  This work is about the complexity behind these proofs: how hard is it to actually find a winning move in a game, when you know by strategy-stealing that one exists?  ...  Chomp is a minimum poset game, where the associated poset holds the squares of the chocolate bar, with the poisoned square removed, and squares are compared by the usual poset relation on Z 2 (note that  ... 
arXiv:1911.06907v1 fatcat:7zfvsscqjfejrdwuqo3skqzbve

Strategy-Stealing Is Non-Constructive

Greg Bodwin, Ofer Grossman, Michael Wagner
2020 Innovations in Theoretical Computer Science  
We prove that this problem is PSPACE-Complete already for Minimum Poset Games and Symmetric Maker-Maker Games, which are simple classes of games that capture two of the main types of strategy-stealing  ...  This work is about the complexity behind these proofs: how hard is it to actually find a winning move in a game, when you know by strategy-stealing that one exists?  ...  Chomp is a minimum poset game, where the associated poset holds the squares of the chocolate bar, with the poisoned square removed, and squares are compared by the usual poset relation on Z 2 (note that  ... 
doi:10.4230/lipics.itcs.2020.21 dblp:conf/innovations/BodwinG20 fatcat:4fnuixrygzff5nhdzdh3fnmk54

FO Model Checking on Posets of Bounded Width [article]

Jakub Gajarský, Petr Hliněný, Daniel Lokshtanov, Jan Obdržálek, Sebastian Ordyniak, M. S. Ramanujan, Saket Saurabh
2015 arXiv   pre-print
Bova, Ganian and Szeider [LICS'14] initiated the study of the complexity of FO model checking on partially ordered sets (posets).  ...  Bova, Ganian and Szeider showed that model checking existential FO logic is fixed-parameter tractable (FPT) on posets of bounded width, where the width of a poset is the size of the largest antichain in  ...  Parameterized Complexity Here we introduce the most basic concepts of parameterized complexity theory. For more details, we refer to the many existing text books on the topic [5, 10, 19 ].  ... 
arXiv:1504.04115v2 fatcat:2bmwaakfync6fkwg2agtkckivy

Page 7848 of Mathematical Reviews Vol. , Issue 2002K [page]

2002 Mathematical Reviews  
The other result concerns the weak order obtained by the contraction of a poset P, that is, by the removal of the transitive closure of the incomparability relation.  ...  As noted by the author, and a consequence of his method of proof, the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups are determined in settling the issue that  ... 

FO Model Checking on Posets of Bounded Width

Jakub Gajarsky, Petr Hlineny, Daniel Lokshtanov, Jan Obdralek, Sebastian Ordyniak, M.S. Ramanujan, Saket Saurabh
2015 2015 IEEE 56th Annual Symposium on Foundations of Computer Science  
Parameterized Complexity Here we introduce the most basic concepts of parameterized complexity theory. For more details, we refer to the many existing text books on the topic [5, 10, 19 ].  ...  In our case the game G(P, ϕ) for a poset P and an FO formula ϕ = ϕ(x 1 , . . . , x k ) in negation normal form (where x 1 , . . . , x k are the free variables of ϕ) is defined as follows: The game is played  ... 
doi:10.1109/focs.2015.63 dblp:conf/focs/GajarskyHLOORS15 fatcat:2h4ymkxotfbnfpplhh6v4aqza4

The Chip Firing Game and Matroid Complexes

Criel Merino
2001 Discrete Models: Combinatorics, Computation, and Geometry  
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.  ...  The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.  ...  The last rule ensures an infinite game. In [Big99a] , the game is called the dollar game, and dollars are used instead of chips.  ... 
dblp:conf/dmccg/Merino01 fatcat:q5fvww5hina2ro7piflxyzpl4i

Eigenstripping, Spectral Decay, and Edge-Expansion on Posets [article]

Jason Gaitonde, Max Hopkins, Tali Kaufman, Shachar Lovett, Ruizhe Zhang
2022 arXiv   pre-print
of the underlying poset.  ...  On the other hand, many important applications (e.g. to locally testable codes, 2-2 games) rely on a more general class of underlying structures called posets, and crucially take advantage of non-simplicial  ...  like the proof of the 2-2 Games Conjecture.  ... 
arXiv:2205.00644v1 fatcat:k4fwheoufbdrtb2roskibf5hr4

Groupoids in combinatorics -- applications of a theory of local symmetries [article]

Rade T. Zivaljevic
2006 arXiv   pre-print
complexes, posets, graphs, polytopes, arrangements and other combinatorial objects.  ...  An objective of the theory of combinatorial groupoids is to introduce concepts like "holonomy", "parallel transport", "bundles", "combinatorial curvature" etc. in the context of simplicial (polyhedral)  ...  This is also a pleasant opportunity to acknowledge the support by the projects no. 144014 and 144026 of the Serbian Ministry of Science and the project "Geometry, Topology and Combinatorics of Manifolds  ... 
arXiv:math/0605508v1 fatcat:qjhqpl2kznb6lehnxhlqepiaua

Computational Complexity of a Solution for Directed Graph Cooperative Games

Ayumi Igarashi, Yoshitsugu Yamamoto
2013 Journal of the Operations Research Society of China  
In this paper, we discuss the computational complexity of the average covering tree value.  ...  We show that computation of the average covering tree value is #P -complete even if the characteristic function of the game is {0, 1}-valued.  ...  Acknowledgements We wish to thank the two anonymous reviewers for their constructive suggestions and comments. The comments have helped us significantly improve the paper.  ... 
doi:10.1007/s40305-013-0025-8 fatcat:yo7jlle3wne6xbsdcyyylmx2nu

Counterexamples to Conjectures About Subset Takeaway and Counting Linear Extensions of a Boolean Lattice

Andries E. Brouwer, J. Daniel Christensen
2017 Order  
We illustrate this algorithm by disproving conjectures about the game Subset Takeaway (Chomp on a hypercube) and computing the number of linear extensions of the lattice of a 7-cube and related lattices  ...  We develop an algorithm for efficiently computing recursively defined functions on posets.  ...  Some positions in these games have rather large Grundy values g. In the table below we give g and the maximal sets of a simplicial complex with this Grundy value, for various positions with n = 7.  ... 
doi:10.1007/s11083-017-9431-6 fatcat:42nx3sp6xzhmvp6szuhnkoujpe
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