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 closing the O(n^3) gap between the two bounds.

In this article, efficient algorithms are presented for the minmax-regret 1-center and 1-median problems on a general graph and a tree with uncertain vertex weights. For the minmax-regret 1-center problem on a general graph, we improve the previous upper bound from O ( mn 2 log n ) to O ( mn log n ). For the problem on a tree, we improve the upper bound from O ( n 2 ) to O ( n log 2 n ). For the minmax-regret 1-median problem on a general graph, we improve the upper bound from O ( mn 2 log n ) to O ( mn 2 + n 3 log n ). For the problem on a tree, we improve the upper bound from O ( n log 2 n ) to O ( n log n ).