# The One-Round Voronoi Game Replayed

## Abstract

We consider the one-round Voronoi game, where player one (“White”, called “Wilma”) places a set of *n* points in a rectangular area of aspect ratio *ρ* ≤ 1, followed by the second player (“Black”, called “Barney”), who places the same number of points. Each player wins the fraction of the board closest to one of his points, and the goal is to win more than half of the total area. This problem has been studied by Cheong et al. who showed that for large enough *n* and *ρ* = 1, Barney has a strategy that guarantees a fraction of 1/2+*α*, for some small fixed *α*.

We resolve a number of open problems raised by that paper. In particular, we give a precise characterization of the outcome of the game for optimal play: We show that Barney has a winning strategy for *n* ≥ 3 and \(\rho > \sqrt{2}/n\), and for *n*=2 and \(\rho > \sqrt{3}/2\). Wilma wins in all remaining cases, i.e., for *n* ≥ 3 and \(\rho \leq \sqrt{2}/n\), for *n*=2 and \(\rho \leq \sqrt{3}/2\), and for *n*=1. We also discuss complexity aspects of the game on more general boards, by proving that for a polygon with holes, it is NP-hard to maximize the area Barney can win against a given set of points by Wilma.

## Keywords

Computational geometry Voronoi diagram Voronoi game Competitive facility location NP-hardness## Preview

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## References

- 1.Ahn, H.-K., Cheng, S.-W., Cheong, O., Golin, M., van Oostrum, R.: Competitive facility location along a highway. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, pp. 237–246. Springer, Heidelberg (2001)CrossRefGoogle Scholar
- 2.Cheong, O., Har-Peled, S., Linial, N., Matousek, J.: The one-round voronoi game. In: Proceedings of the Eighteenth Annual ACM Symposium on Computational Geometry, pp. 97–101 (2002)Google Scholar
- 3.Dehne, F., Klein, R., Seidel, R.: Maximizing a Voronoi region: The convex case. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 624–634. Springer, Heidelberg (2001)Google Scholar
- 4.Eiselt, H., Laporte, G.: Competitive spatial models. European Journal of Operational Research 39, 231–242 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
- 5.Eiselt, H., Laporte, G., Thisse, J.-F.: Competitive location models: A framework and bibliography. Transportation Science 27, 44–54 (1993)zbMATHCrossRefGoogle Scholar
- 6.S. P. Fekete, J. S. B.Mitchell, and K. Weinbrecht. On the continuous Weber and
*k*-median problems. In Proceedings of the Sixteenth Annual ACM Symposium on Computational Geometry, pages 70–79, 2000. Google Scholar - 7.T. Friesz, R. Tobin, and T. Miller. Existence theory for spatially competitive network facility location models. Annals of Operations Research, 18:267–276, 1989. Google Scholar
- 8.D. Lichtenstein. Planar formulae and their uses. SIAM Journal on Computing, 11, 2:329–343, 1982. Google Scholar
- 9.D. Lichtenstein and M. Sipser. Go is polynomial-space hard. Journal of theACM, 27:393–401, 1980. Google Scholar
- 10.J. Robson. The complexity of go. In Information Processing: Proceedings of IFIP Congerss, pages 413–4417, 1983. Google Scholar
- 11.P. Rosenstiehl and R. E. Tarjan. Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete and Computational Geometry, 1:343–353, 1986.Google Scholar