Finding Optimal Bipartitions of Points and Polygons

We give efficient algorithms to compute an optimal bipartition of a set of points or a set of simple polygons in the plane. In particular, we give an O(n 2 ) algorithm for partitioning a set of n points into two subsets in order to minimize the sum of the perimeters of the convex hulls. We also examine the problems of minimizing the maximum of the perimeters, or the sum/maximum of the areas, for the case in which the partitioning is a line partitioning, induced by some line, and we examine related problems for the bipartitioning of polygons. 1 Introduction In the "bipartition problem", we are interested in partitioning a set S of n points into two subsets (S 1 and S 2 ) in such a way as to optimize some function of the "sizes" (¯(S i )) of the two subsets. Avis ([2]) gave an O(n 2 log n) time algorithm to find a bipartition that minimizes the maximum of the diameters of the sets S 1 and S 2 . Asano, Bhattacharya, Keil and Yao ([1]) improved the bound on the time complexity of t..