Voronoi Game on Graphs

TL;DR

This paper studies a Voronoi game on graphs, providing a polynomial-time algorithm for trees, proving NP-completeness on general graphs, and proposing an approximation algorithm for facility placement.

Contribution

It introduces a dynamic programming approach for the game on trees, proves NP-completeness on general graphs, and offers a $1 - 1/e$ approximation algorithm.

Findings

Polynomial-time algorithm for trees

NP-completeness on general graphs

Approximation algorithm with factor $1 - 1/e$

Abstract

\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 $e \approx 2.718$.