Document # On One-Round Discrete Voronoi Games

### Authors Mark de Berg, Sándor Kisfaludi-Bak, Mehran Mehr ## File

LIPIcs.ISAAC.2019.37.pdf
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## Cite As

Mark de Berg, Sándor Kisfaludi-Bak, and Mehran Mehr. On One-Round Discrete Voronoi Games. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 37:1-37:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.37

## Abstract

Let V be a multiset of n points in R^d, which we call voters, and let k >=slant 1 and l >=slant 1 be two given constants. We consider the following game, where two players P and Q compete over the voters in V: First, player P selects a set P of k points in R^d, and then player Q selects a set Q of l points in R^d. Player P wins a voter v in V iff dist(v,P) <=slant dist(v,Q), where dist(v,P) := min_{p in P} dist(v,p) and dist(v,Q) is defined similarly. Player P wins the game if he wins at least half the voters. The algorithmic problem we study is the following: given V, k, and l, how efficiently can we decide if player P has a winning strategy, that is, if P can select his k points such that he wins the game no matter where Q places her points. Banik et al. devised a singly-exponential algorithm for the game in R^1, for the case k=l. We improve their result by presenting the first polynomial-time algorithm for the game in R^1. Our algorithm can handle arbitrary values of k and l. We also show that if d >= 2, deciding if player P has a winning strategy is Sigma_2^P-hard when k and l are part of the input. Finally, we prove that for any dimension d, the problem is contained in the complexity class exists for all R, and we give an algorithm that works in polynomial time for fixed k and l.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
• Theory of computation → Algorithmic game theory
• Theory of computation → Problems, reductions and completeness
##### Keywords
• competitive facility location
• plurality point

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## References

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