Voronoi game on graphs

Abstract

Voronoi game is a geometric model of competitive facility location problem, where each market player comes up with a set of possible locations for placing their facilities. The objective of each player is to maximize the region occupied on the underlying space. In this paper we consider one round Voronoi game with two players. Here the underlying space is a road network, which is modeled by a graph embedded on ℝ2. In this game each of the players places a set of facilities and the underlying graph is subdivided according to the nearest neighbor rule. The player which dominates the maximum region of the graph wins. Given a placement of facilities by Player 1, we have characterized the optimal placement by Player 2. At first we dealt with the case when Player 2 places a constant number of facilities and provided an algorithm for the same. Next we have proved that finding the optimal placement of k facilities by Player 2 is NP-hard where k is given. Lastly we presented a 1.58 factor approximation algorithm for the above mentioned problem. © 2013 Springer-Verlag.