The $(1|1)_R$-Centroid Problem on the Plane

TL;DR

This paper introduces a more efficient algorithm for the $(1|1)_R$-centroid problem on the plane, reducing the complexity from $O(n^5 ext{ log } n)$ to $O(n^2 ext{ log } n)$ by using a new parametric search approach.

Contribution

It presents a novel $O(n^2 ext{ log } n)$-time algorithm for the general $(1|1)_R$-centroid problem, improving upon the previous $O(n^5 ext{ log } n)$ solution.

Findings

Reduced algorithm complexity from $O(n^5 ext{ log } n)$ to $O(n^2 ext{ log } n)$

Identified $O(n^4)$ candidate points for optimal solution

Close the computational gap between the two problem versions

Abstract

In 1982, Drezner proposed the (1|1)-centroid problem on the plane, in which two players, called the leader and the follower, open facilities to provide service to customers in a competitive manner. The leader opens the first facility, and then the follower opens the second. Each customer will patronize the facility closest to him (ties broken in favor of the leader's one), thereby decides the market share of the two players. The goal is to find the best position for the leader's facility so that his market share is maximized. The best algorithm for this problem is an $O(n^2 \log n)$-time parametric search approach, which searches over the space of possible market share values. In the same paper, Drezner also proposed a general version of (1|1)-centroid problem by introducing a minimal distance constraint $R$, such that the follower's facility is not allowed to be located within a distance $R$ from the leader's. He proposed an $O(n^5 \log n)$-time algorithm for this general version by identifying $O(n^4)$ points as the candidates of the optimal solution and checking the market share for each of them. In this paper, we develop a new parametric search approach searching over the $O(n^4)$ candidate points, and present an $O(n^2 \log n)$-time algorithm for the general version, thereby close the $O(n^3)$ gap between the two bounds.