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Time-Optimal Winning Strategies for Poset Games [chapter]

Martin Zimmermann
2009 Lecture Notes in Computer Science  
We show that optimal strategies (with respect to long term average accumulated waiting times) exist.  ...  We introduce a novel winning condition for infinite two-player games on graphs which extends the request-response condition and better matches concrete applications in scheduling or project planning.  ...  I want to thank him for his advice and suggestions. Also, I want to thank the anonymous referees of [8] for their helpful remarks.  ... 
doi:10.1007/978-3-642-02979-0_25 fatcat:kx3qaoejsndydizasrcskawjsm

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  ...  In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing.  ...  to 2 . • Output: any optimal first move for the first player in the associated game.  ... 
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  ...  In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing.  ...  Theorem 2 generalizes to show that any minimum poset game is a win for the first player.  ... 
doi:10.4230/lipics.itcs.2020.21 dblp:conf/innovations/BodwinG20 fatcat:4fnuixrygzff5nhdzdh3fnmk54

Chain-making games in grid-like posets

Daniel W. Cranston, William B. Kinnersley, Kevin G. Milans, Gregory J. Puleo, Douglas B. West
2012 Journal of Combinatorics  
We study the Maker-Breaker game on the hypergraph of chains of fixed size in a poset.  ...  game.  ...  To show optimality of the resulting strategy for Maker, we present a strategy for Breaker.  ... 
doi:10.4310/joc.2012.v3.n4.a3 fatcat:horpht2prfbulergkf36blq4aq

Chain-making games in grid-like posets [article]

Daniel W. Cranston, William B. Kinnersley, Kevin G. Milans, Gregory J. Puleo, Douglas B. West
2011 arXiv   pre-print
We study the Maker-Breaker game on the hypergraph of chains of fixed size in a poset.  ...  game.  ...  To show optimality of the resulting strategy for Maker, we present a strategy for Breaker.  ... 
arXiv:1108.0710v1 fatcat:vzhkzci3fbenjn4kj5gixolkzu

Asymmetric Coloring Games on Incomparability Graphs

Tomasz Krawczyk, Bartosz Walczak
2015 Electronic Notes in Discrete Mathematics  
The game chromatic number of G is the minimum number of colors that allows Alice to win the game.  ...  In this extended abstract, we study the behavior of these parameters for incomparability graphs of posets with bounded width.  ...  The game chromatic number of G, denoted by χ g (G), is the minimum number of colors for which Alice has a winning strategy in the coloring game on G.  ... 
doi:10.1016/j.endm.2015.06.108 fatcat:lhwz25z73jgefeaj65jostre3e

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

Stephen A. Fenner, John Rogers
2015 arXiv   pre-print
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.  ...  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.  ...  We say that a poset P is an ∃-game (or winning position) if the first player has a winning strategy, and P is a ∀game (or losing position) if the second player has a winning strategy.  ... 
arXiv:1505.07416v2 fatcat:yxcmoncmpvailp27irc7e3ydbi

Asymmetric coloring games on incomparability graphs [article]

Tomasz Krawczyk, Bartosz Walczak
2015 arXiv   pre-print
We study the behavior of these parameters for incomparability graphs of posets with bounded width.  ...  The game chromatic number of G is the minimum number of colors that allows Alice to win the game.  ...  The game chromatic number of G, denoted by χ g (G), is the minimum number of colors for which Alice has a winning strategy in the coloring game on G.  ... 
arXiv:1503.04748v1 fatcat:kqmmyg4ndzd2bc3sgefq32ewru

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

Andries E. Brouwer, J. Daniel Christensen
2017 Order  
We develop an algorithm for efficiently computing recursively defined functions on posets.  ...  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  ...  The first is that of finding the optimal strategy for the game of Chomp (or Subset Takeaway). The second is that of counting the number of linear extensions.  ... 
doi:10.1007/s11083-017-9431-6 fatcat:42nx3sp6xzhmvp6szuhnkoujpe

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

Erik Järleberg, Jonas Sjöstrand
2014 arXiv   pre-print
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, 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.  ...  So, if Black starts playing the game G − G L − 1 we must show that White has a winning strategy.  ... 
arXiv:1405.1914v1 fatcat:jagpx53c6jh25jmaat5vuac64u

Numerical Tic-Tac-Toe on the 4×4 Board [chapter]

Bryce Sandlund, Kerrick Staley, Michael Dixon, Steve Butler
2014 Lecture Notes in Computer Science  
In this paper we consider the 4 × 4 game and determine that the second player has the advantage.  ...  The original 3 × 3 game was created and analyzed by Ron Graham nearly fifty years ago and has been shown that the first player has the advantage.  ...  By using an appropriately pruned tree for the question "Can player two force a win?" we would have a perfect strategy for the game.  ... 
doi:10.1007/978-3-319-08783-2_46 fatcat:fyc4jhrahbeelasxnx2uihpod4

Small Resolution Proofs for QBF using Dependency Treewidth [article]

Eduard Eiben, Robert Ganian, Sebastian Ordyniak
2017 arXiv   pre-print
Dependency treewidth pushes the frontiers of tractability for QBF by overcoming the limitations of previously introduced variants of treewidth for QBF.  ...  This is especially true for the structural parameter treewidth, which has allowed the design of successful algorithms for SAT but cannot be straightforwardly applied to QBF since it does not take into  ...  The authors wish to thank the anonymous reviewers for their helpful comments. Eduard Eiben acknowledges support by the Austrian Science Fund (FWF, projects P26696 and W1255-N23).  ... 
arXiv:1711.02120v1 fatcat:dvyygn3dyrdudn7hqchxyg53j4

Page 2072 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
A master for one of these games is an agent who plays a winning strategy.  ...  However, if in addition to getting a program, a machine may also watch masters play winning strategies, then the machine is able to incrementally learn a winning strategy for the given game.  ... 

Complexity, appeal and challenges of combinatorial games

Aviezri S. Fraenkel
2004 Theoretical Computer Science  
For algorithmicians, such studies provide new interesting algorithmic challenges.  ...  Substantiations of these assertions are illustrated on hand of many sample games, leading to a deÿnition of the tractability, polynomiality and e ciency of subsets of games.  ...  Yet, for annihilation games, the only strategy that we know which can produce a next winning move from an N -position in polynomial time, is a strategy in the broad sense.  ... 
doi:10.1016/j.tcs.2002.11.001 fatcat:iay4nt562re2xniyhtt6l2p2cy

Isolating Cardinal Invariants [article]

Jindrich Zapletal
2001 arXiv   pre-print
There is an optimal way to increase certain cardinal invariants of the continuum.  ...  This is obviously a winning strategy for Adam.  ...  For the left to right direction fix a winning strategy σ for Adam.  ... 
arXiv:math/0106202v1 fatcat:rmfsrf2bdjhobkxec35j3fcwqi
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