图上的Voronoi对策
Voronoi Game on Graphs
摘要：
\textit{Voronoi game}是两个参与者之间的竞争设施选址问题的几何模型。用户通常被建模为均匀分布在给定的底层空间上的点。每个玩家在底层空间中选择一组点来放置他们的设施。每个用户都可以从最近的设施获得服务。设施的服务区由比任何其他设施更接近它的用户集合组成。每个玩家的收益由其所有设施服务的用户数量定义。每个玩家的目标是最大化他们各自的收益。在本文中，我们考虑两个球员{ \it Voronoi游戏}的基础空间是一个由图建模的道路网络。在这个框架中，我们考虑的问题，找到$k$最佳设施位置的球员2给定任何位置的$m$设施的球员1。我们的主要结果是一个动态规划为基础的多项式时间算法的树网络上的这个问题。另一方面，我们证明了这个问题是强$\mathcal{NP}$ -完全的图。这证明了寻找P2的获胜策略是$\mathcal{NP}$ -完全的。因此，我们设计了一个$1-\frac{1}{e}$因子近似算法，其中$e \approx 2.718$。
英文摘要：
\textit{Voronoi game} is a geometric model of competitive facility location problem played between two players. Users are generally modeled as points uniformly distributed on a given underlying space. Each player chooses a set of points in the underlying space to place their facilities. Each user avails service from its nearest facility. Service zone of a facility consists of the set of users which are closer to it than any other facility. Payoff of each player is defined by the quantity of users served by all of its facilities. The objective of each player is to maximize their respective payoff. In this paper we consider the two players {\it Voronoi game} where the underlying space is a road network modeled by a graph. In this framework we consider the problem of finding $k$ optimal facility locations of Player 2 given any placement of $m$ facilities by Player 1. Our main result is a dynamic programming based polynomial time algorithm for this problem on tree network. On the other hand, we show that the problem is strongly $\mathcal{NP}$-complete for graphs. This proves that finding a winning strategy of P2 is $\mathcal{NP}$-complete. Consequently, we design an $1-\frac{1}{e}$ factor approximation algorithm, where $e \approx 2.718$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
