The Complexity of Node Blocking for Dags

📝 Abstract

We consider the following modification of annihilation game called node blocking. Given a directed graph, each vertex can be occupied by at most one token. There are two types of tokens, each player can move his type of tokens. The players alternate their moves and the current player $i$ selects one token of type $i$ and moves the token along a directed edge to an unoccupied vertex. If a player cannot make a move then he loses. We consider the problem of determining the complexity of the game: given an arbitrary configuration of tokens in a directed acyclic graph, does the current player has a winning strategy? We prove that the problem is PSPACE-complete.

💡 Analysis

We consider the following modification of annihilation game called node blocking. Given a directed graph, each vertex can be occupied by at most one token. There are two types of tokens, each player can move his type of tokens. The players alternate their moves and the current player $i$ selects one token of type $i$ and moves the token along a directed edge to an unoccupied vertex. If a player cannot make a move then he loses. We consider the problem of determining the complexity of the game: given an arbitrary configuration of tokens in a directed acyclic graph, does the current player has a winning strategy? We prove that the problem is PSPACE-complete.

📄 Content

arXiv:0802.3513v1 [cs.GT] 24 Feb 2008 The Complexity of Node Blocking for Dags Dariusz Dereniowski Department of Algorithms and System Modeling, Gda´nsk University of Technology, Poland deren@eti.pg.gda.pl November 10, 2018 Abstract: We consider the following modification of annihilation game called node blocking. Given a directed graph, each vertex can be occu- pied by at most one token. There are two types of tokens, each player can move his type of tokens. The players alternate their moves and the current player i selects one token of type i and moves the token along a directed edge to an unoccupied vertex. If a player cannot make a move then he loses. We consider the problem of determining the complexity of the game: given an arbitrary configuration of tokens in a directed acyclic graph, does the current player has a winning strategy? We prove that the problem is PSPACE-complete. Keywords: annihilation game, node blocking, PSPACE-completeness 1 Introduction The study of annihilation games has been suggested by John Conway and the first papers were published by Fraenkel and Yesha [7, 9]. They considered a 2-player game played on an underlying directed graph G (possibly with cycles). The current player selects a token and moves it along an arc outgoing from a vertex containing the token. If a vertex contains two tokens then they are removed from G (annihilation). Authors in [9] gave a polynomial-time algorithm for computing a winning strategy. In this paper, including all the mentioned here results, we assume the normal play, where the first player unable to make a move loses (mis`ere annihilation games have been considered in [2]). Fraenkel considered in [4] a generalization of cellular-automata games to two- player games and provided a strategy for such cases. In particular, if for each vertex there is at most one outgoing arc then it is possible to derive a polynomial-time strategy [4]. Since the formulation of the game is equivalent to the one mentioned above, this result can be directly applied for the annihilation game. Fraenkel in [3] studied the connections between annihilation games and error- correcting codes. The authors in [6] gave an algorithm for computing error-correcting 1 codes. The algorithm is polynomial in the size of the code and uses the theory of two-player cellular-automata games. In the following we are interested in generalizations of the annihilation game, where there is more than one type of token and/or there is a different interaction be- tween the tokens. Assume that r ≥2 types of tokens are given and each type of token can be moved along a subset of the edges. Given a configuration of tokens in a graph, deciding whether the current player has a winning strategy is PSPACE-complete for acyclic graphs [5]. A modification called hit, where r ≥2 types of tokens and edges are distinguished was considered in [5]. A move consists of selecting a token of type i and moving along an arc of type i ∈{1, . . ., r}. The target vertex v cannot be occupied by a token of type i, but if v contains token of other type then it is removed (so, when the move ends v is occupied by the token of type i). The complexity of determining the outcome of this game is PSPACE-complete for acyclic graphs and r = 2 [5]. A modification of hit called capture has the same rules except that each token can travel along any edge. Capture is PSPACE-complete for acyclic and EXPTIME-complete for general graphs [10]. In a node blocking each token is of one of the two types. Each vertex can contain at most one token. Player i can move the tokens of type i, i = 1, 2. All tokens can move along all arcs. A player i makes a move, by selecting one token of type i (occupying a vertex v ∈V) and an unoccupied vertex u ∈V such that (v, u) ∈E and moving the token from v to u. The first player unable to make a move loses and his opponent wins the game. There is a tie if there is no last move. First, the game was proved to be NP-hard [8], then PSPACE-hard for general graphs [5]. The complexity for general graphs has been finally proved in [10] to be EXPTIME-complete. In an edge blocking all tokens are identical, i.e. each player can move any token, while each arc is of type 1 or 2 and a player i makes his move by moving a token along an arc of type i, i = 1, 2. Similarly as before, the first player who cannot make a move loses. A tie occurs if there is no last move. This game is PSPACE-complete for dags. The following table summarizes the complexity of all the mentioned two-player annihilation games. We list only the strongest known results. Game: dag general Annihilation PSPACE-complete [5] ?∗ Hit PSPACE-complete [5] ?∗ Capture PSPACE-complete [10] EXPTIME-complete [10] Node blocking ? EXPTIME-complete [10] Edge blocking PSPACE-complete [5] ?∗ Note that for the entries labeled as “?∗” can be replaced by “PSPACE-hard” (which can be concluded from the corresponding results for acyclic graphs), but the question re- mains whether the games are

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