Phutball is PSPACE-hard

📝 Abstract

We consider the $n\times n$ game of Phutball. It is shown that, given an arbitrary position of stones on the board, it is a PSPACE-hard problem to determine whether the specified player can win the game, regardless of the opponent’s choices made during the game.

💡 Analysis

We consider the $n\times n$ game of Phutball. It is shown that, given an arbitrary position of stones on the board, it is a PSPACE-hard problem to determine whether the specified player can win the game, regardless of the opponent’s choices made during the game.

📄 Content

arXiv:0804.1777v2 [cs.GT] 20 Mar 2009 Phutball is PSPACE-hard Dariusz Dereniowski Department of Algorithms and System Modeling, Gda´nsk University of Technology, Poland deren@eti.pg.gda.pl October 23, 2018 Abstract: We consider the n × n game of Phutball. It is shown that, given an arbitrary position of stones on the board, it is a PSPACE-hard problem to determine whether the specified player can win the game, regardless of the op- ponent’s choices made during the game. Keywords: computational complexity, games, Phutball, Pspace hardness 1 Introduction There is a deep mathematical theory developed for analyzing combinatorial games [1, 8]. The researchers work on the algorithmic techniques which are useful for finding good game strategies for many board games, including Phutball [4, 5]. The paper [6] introduces the notion of generalized threads and this technique is used to solve some Go instances and the author suggest that this approach could be effective for other board games, like Phutball. T.Cazenave used an approach called Gradual Abstract Proof Search to show that 11×11 Phutball is a win for the first player [7]. The game is loopy, i.e. it is possible to obtain a configuration of stones which already appeared in one of the previous turns – some combinatorial aspects of loopy games were considered in [24]. In this paper we are interested in the complexity of the game rather than in manipulating and analyzing the rooted tree describing the game. Several generalizations of one-player games turns out 1 to be NP-complete: Peg Solitaire [26], Minesweeper (the problem of testing con- sistency) [20], Same Game [3]. However, most of the board games (especially two-player games) appear to be harder: Checkers [23], Hex (a generalization to graphs) [13], Othello (Reversi) [19], Sokoban [9], Go [21, 22, 27], Dyson Tele- scopes [12], Rush Hour [14] or Amazons [16]. The Phutball [2] game is usually played on a 19 × 19 Go board. Initially a black stone is placed in the middle of the board. The players make their moves alternately. A player makes his move by either placing a white stone in an unoccu- pied position, or makes a sequence of jumps over horizontal, vertical or diagonal sequences of white stones. Each jump is performed by moving the black stone, called ball, over a line of white stones (no empty space between the ball and the line is allowed if we want to make a jump) and placing the ball on the board on the first unoccupied position after the last white stone in the line. The white stones are removed from the board immediately after the jump. Each player tries to move the ball on or over the opponent’s goal line. The goal lines are two opposite edges of the board. We consider a natural generalization with an arbitrary size of the board and initially a black stone placed in the middle of the board. As indicated in [2], Phutball is not the kind of game where you can expect a complete analysis. The authors in [17] considered a simplified version of the game, i.e. the case where there is only one dimension and it turns out that ac- cording to the presented examples, the one-dimensional version still seems to be hard to analyze. Moreover, given an arbitrary position in the 2-dimensional Phut- ball game, it is an NP-complete problem to determine whether the current player can win the game in his next move [11]. However, as indicated in several papers [7, 10, 11, 18], the complexity of the Phutball game is still open. In this paper we place the problem of determining whether the current player has a winning strategy in the class of PSPACE-hard problems. 2 A graph game We start this section by describing the rules of a game played on a graph. Then we prove that this game is PSPACE-hard. The graph constructed on the basis of a problem known to be PSPACE-complete is defined in such a way that its topology allows to code it as a configuration of stones in the Phutball game. The game described in the following is played on a directed graph. For com- pleteness we list here some basic definitions. A directed graph G is a pair G = (V(G), E(G)) with a vertex set V(G) and a set of directed edges E(G) (each e ∈ 2 E(G) is an ordered pair of two vertices). We say that H is a subgraph of G, H ⊆G, if V(H) ⊆V(G) and E(H) ⊆E(G). A directed path P = ({v1, . . . , vn}, E(P)) from v1 to vn is a graph with edge set E(P) = {(vi, vi+1) : i = 1, . . ., n −1}. The vertices V(P) \ {v1, vn} are the internal vertices of P. The input of the game is a directed graph G = (V(G), E(G)), a set C ⊆V(G), a vertex s ∈C, and a relation R ⊆V(G) × E(G) between the vertices and the edges of G. If (v, e) ∈R then we say that a vertex v is pointing an edge e. Denote by R−1(E(G)) the set of vertices v for which there exists e ∈E(G), such that (v, e) ∈R. The players of the game will be called ∃-player and ∀-player. We will also use a notation that if a symbol X refers to one of the players then X is the other player. At each point of the game there is a unique activ

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