Existence theory for spatially competitive network facility location models
Annals of Operations Research
Competitive facility location: the Voronoi game
Theoretical Computer Science
Finding a Guard that Sees Most and a Shop that Sells Most
Discrete & Computational Geometry
(r,p)-centroid problems on paths and trees
Theoretical Computer Science
The one-round Voronoi game replayed
Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
Journal of Mathematical Modelling and Algorithms
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The one-round discrete Voronoi game, with respect to a n-point user set $\mathcal {U}$ , consists of two players Player 1 (P1) and Player 2 (P2). At first, P1 chooses a set $\mathcal{F}_{1}$ of m facilities following which P2 chooses another set $\mathcal{F}_{2}$ of m facilities, disjoint from $\mathcal{F}_{1}$ , where m(=O(1)) is a positive constant. The payoff of P2 is defined as the cardinality of the set of points in $\mathcal{U}$ which are closer to a facility in $\mathcal{F}_{2}$ than to every facility in $\mathcal{F}_{1}$ , and the payoff of P1 is the difference between the number of users in $\mathcal{U}$ and the payoff of P2. The objective of both the players in the game is to maximize their respective payoffs. In this paper, we address the case where the points in $\mathcal{U}$ are located along a line. We show that if the sorted order of the points in $\mathcal{U}$ along the line is known, then the optimal strategy of P2, given any placement of facilities by P1, can be computed in O(n) time. We then prove that for m驴2 the optimal strategy of P1 in the one-round discrete Voronoi game, with the users on a line, can be computed in $O(n^{m-\lambda_{m}})$ time, where 0驴 m m.