Are there more almost separable partitions than separable partitions?

Abstract

A partition of a set of n points in d-dimensional space into p parts is called an (almost) separable partition if the convex hulls formed by the parts are (almost) pairwise disjoint. These two partition classes are the most encountered ones in clustering and other partition problems for high-dimensional points and their usefulness depends critically on the issue whether their numbers are small. The problem of bounding separable partitions has been well studied in the literature (Alon and Onn in Discrete Appl. Math. 91:39-51, 1999; Barnes et al. in Math. Program. 54:69-86, 1992; Harding in Proc. Edinb. Math. Soc. 15:285-289, 1967; Hwang et al. in SIAM J. Optim. 10:70-81, 1999; Hwang and Rothblum in J. Comb. Optim. 21:423-433, 2011a). In this paper, we prove that for da parts per thousand currency sign2 or pa parts per thousand currency sign2, the maximum number of almost separable partitions is equal to the maximum number of separable partitions.