Geography

Fraenkel, Aviezri S.; Simonson, Shai

doi:10.1016/0304-3975(93)90356-x

Cited by 18 publications

(11 citation statements)

References 8 publications

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“…For example, Node Kayles is proved to be polynomial on paths and cographs [4]. This is also the case for Geography played on undirected graphs [19]. Finally, note that the complexity of Domineering is still an open problem.…”

Section: Complexity Of Combinatorial Gamesmentioning

confidence: 86%

Combinatorial Games: From Theoretical Solving to AI Algorithms

Duchêne

2016

Lecture Notes in Computer Science

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Combinatorial game solving is a research field that is frequently highlighted each time a program defeats the best human player: Deep Blue (IBM) vs Kasparov for Chess in 1997, and Alpha Go (Google) vs Lee Sedol for the game of Go in 2016. But what is hidden behind these success stories ? First of all, I will consider combinatorial games from a theoretical point of view. We will see how to proceed to properly define and deal with the concepts of outcome, value, and winning strategy. Are there some games for which an exact winning strategy can be expected? Unfortunately, the answer is no in many cases (including some of the most famous ones like Go, Othello, Chess or Checkers), as exact game solving belongs to the problems of highest complexity. Therefore, finding out an effective approximate strategy has highly motivated the community of AI researchers. In the current survey, the basics of the best AI programs will be presented, and in particular the well-known Minimax and Monte-Carlo Tree Search approaches.

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Section: Complexity Of Combinatorial Gamesmentioning

confidence: 86%

Combinatorial Games: From Theoretical Solving to AI Algorithms

Duchêne

2016

Lecture Notes in Computer Science

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“…This was done as an intermediate step for showing that Go is PSPACE-complete. Fraenkel and Simonson [9] gave polynomial time algorithms for Geography and Vertex Geography when played on directed acyclic graphs. In Fraenkel et al [10] it was proved that also the undirected variant of Geography is PSPACE-complete.…”

Section: Related Literaturementioning

confidence: 99%

On the Shortest Path Game

Darmann

Pferschy

Schauer

2017

Discrete Applied Mathematics

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In this work we address a game theoretic variant of the shortest path problem, in which two decision makers (players) move together along the edges of a graph from a given starting vertex to a given destination. The two players take turns in deciding in each vertex which edge to traverse next. The decider in each vertex also has to pay the cost of the chosen edge. We want to determine the path where each player minimizes its costs taking into account that also the other player acts in a selfish and rational way. Such a solution is a subgame perfect equilibrium and can be determined by backward induction in the game tree of the associated finite game in extensive form.We show that the decision problem associated with such a path is PSPACEcomplete even for bipartite graphs both for the directed and the undirected version. The latter result is a surprising deviation from the complexity status of the closely related game Geography.On the other hand, we can give polynomial time algorithms for directed acyclic graphs and for cactus graphs even in the undirected case. The latter is based on a decomposition of the graph into components and their resolution by a number of fairly involved dynamic programming arrays. Finally, we give some arguments about closing the gap of the complexity status for graphs of bounded treewidth.

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“…Figure 3 illustrates the computation of the lo function. Note that one of the reasons for which we have slightly changed Stockman's rules is that we assumed that our current game had a lower complexity than Vertex NimG or Geography [14] on directed graphs. The previous theorem shows that our assumption was somehow true, since we remind the reader that Vertex NimG was proved to be PSPACE-hard with all loops (for the move then remove convention [4], the other convention being trivial with all loops).…”

Section: Directed Graphs With All Loopsmentioning

confidence: 99%

Vertex Nim played on graphs

Duchêne

Renault

2014

Theoretical Computer Science

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Abstract. Given a graph G with positive integer weights on the vertices, and a token placed on some current vertex u, two players alternately remove a positive integer weight from u and then move the token to a new current vertex adjacent to u. When the weight of a vertex is set to 0, it is removed and its neighborhood becomes a clique. The player making the last move wins. This adaptation of Nim on graphs is called Vertexnim, and slightly differs from the game Vertex NimG introduced by Stockman in 2004. Vertexnim can be played on both directed or undirected graphs. In this paper, we study the complexity of deciding whether a given game position of Vertexnim is winning for the first or second player. In particular, we show that for undirected graphs, this problem can be solved in quadratic time. Our algorithm is also available for the game Vertex NimG, thus improving Stockman's exptime algorithm. In the directed case, we are able to compute the winning strategy in polynomial time for several instances, including circuits or digraphs with self loops.

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Geography

Cited by 18 publications

References 8 publications

Combinatorial Games: From Theoretical Solving to AI Algorithms

Combinatorial Games: From Theoretical Solving to AI Algorithms

On the Shortest Path Game

Vertex Nim played on graphs

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