The Voronoi game on graphs and its complexity

The Voronoi game is a two-person game which is a model for a competitive facility location. The game is played on a continuous domain, and only two special cases (one-dimensional case and one-round case) are well investigated. We introduce the discrete Voronoi game in which the game arena is given as a graph. We first analyze the game when the arena is a large complete k-ary tree, and give an optimal strategy. When both players play optimally, the first player wins when k is odd, and the game ends in a tie for even k. Next we show that the discrete Voronoi game is intractable in general. Even for the one-round case in which the strategy adopted by the first player consist of a fixed single node, deciding whether the second player can win is NP-complete. We also show that deciding whether the second player can win is PSPACE-complete in general. Submitted: May 2009 Reviewed: October 2009 Revised: July 2011 Accepted: August 2011 Final: August 2011 Published: August 2011 Article type: Regular Paper Communicated by: H. Meijer A preliminary version was presented at the 2nd IEEE Symposium on Computational Intelligence and Games (CIG 2006). A part of this work was done while the third author was