Voronoi game on polygons

Abstract

The competitive facility location problem is the problem of determining facility locations involving multiple players to optimize their various gains. The Voronoi game is a competitive facility location problem on a given arena played by two players, the server and the adversary. The players alternately take turns, one or more times, to place their facilities in the arena with a predetermined set of n clients, where both facilities and clients are denoted by points, to maximize some resource gain. The Voronoi game on a polygonP is a type of competitive facility location problem where n clients are located on the boundary of P. The server, Alice, and adversary, Bob, are respectively in the interior and the exterior of the polygon P at locations A and B, respectively. Additionally, the metrics for Alice and Bob are the internal and external geodesic distances for the polygon P, respectively. In this paper, we present some surprising results on the Voronoi games on polygons.We prove lower and upper bounds of ⌈n/3⌉ and n−1 respectively in the single-round game for the number of clients won by the server for n clients. Both bounds are tight. In the process, we show that in some convex polygons, the adversary wins no more than k clients in a k-round Voronoi game for any k≤n. Consequentially, the adversary Bob does not have a guaranteed good winning strategy even for the simpler case of convex polygons, i.e., there exist convex polygons such that no placement of B guarantees more than k clients in the k-round game. We also design O(nlog2⁡n+mlog⁡n) and O(n+m) time algorithms to compute the optimal locations for the server and the adversary respectively to maximize their client counts where the convex polygon has size m. Moreover, we present an O(nlog⁡n) time algorithm to compute the common intersection of a set of n ellipses. This is needed in our algorithm and may be of independent interest.Lastly, we present some results on the Voronoi games, where the arena is a convex polytope. The server and adversary are respectively in the interior and exterior of P, and the clients are on the polytope boundary.