Discrete Voronoi Games and $\epsilon$-Nets, in Two and Three Dimensions

The one-round discrete Voronoi game, with respect to a n-point user set U , consists of two players Player 1 (P1) and Player 2 (P2). At first, P1 chooses a set of facilities F1 following which P2 chooses another set of facilities F2, disjoint from F1. The payoff of P2 is defined as the cardinality of the set of points in U which are closer to a facility in F2 than to every facility in F1, and the payoff of P1 is the difference between the number of users in U and the payoff of P2. The objective of both the players in the game is to maximize their respective payoffs. In this paper we study the one-round discrete Voronoi game where P1 places k facilities and P2 places one facility and we have denoted this game as V G(k, 1). Although the optimal solution of this game can be found in polynomial time, the polynomial has a very high degree. In this paper, we focus on achieving approximate solutions to V G(k, 1) with significantly better running times. We provide a constant-factor approximate solution to the optimal strategy of P1 in V G(k, 1) by establishing a connection between V G(k, 1) and weak -nets. To the best of our knowledge, this is the first time that Voronoi games are studied from the point of view of -nets.