Sachio Teramoto

Voronoi game on graphs and its complexity

The Voronoi game is a two-person game which is a model for competitive facility location. The game is played on a continuous domain, and only two special cases (the 1D case and the 1-round case) are well investigated. We introduce the discrete Voronoi game in which the game arena is given as a graph. We first show the best strategy when the game arena is a large complete k-ary tree. Next, we show that the discrete Voronoi game is intractable in general. Even in the 1-round case, and the place occupied by the first player is fixed, the game is NP-complete in general. We also show that the game is PSPACE-complete in general case

Inserting Points Uniformly at Every Instance

Arranging n points as uniformly as possible is a frequently occurring problem. It is equivalent to packing n equal and non-overlapping circles in a unit square. In this paper we generalize this problem in such a way that points are inserted one by one with uniformity preserved at every instance. Our criterion for uniformity is to minimize the gap ratio (which is the maximum gap over the minimum gap) at every point insertion. We present a linear time algorithm for finding an optimal n-point sequence with the maximum gap ratio bounded by 2⌊n/2⌋/(⌊n/2⌋+1) in the 1-dimensional case. We describe how hard the same problem is for a point set in the plane and propose a local search heuristics for finding a good solution.