The Combinatorics of Splitting and Splittable Families

A set A is said to split a finite set B if exactly half the elements of B (up to rounding) are contained in A. We study the dual notions: (1) a splitting family is a collection of sets such that any subset of {1,2,...,k} is split by a set in the family, and (2) a splittable family is a collection of sets such that there is a single set A that splits each set in the family. Splitting families were introduced to optimize combinatorial search algorithms, while splittable families arise naturally in applications of discrepancy theory, such as parallelization and halftoning. In this presentation, we present improved bounds on the minimum size of a splitting family. We also give results on minimally splittable families, i.e., families that have the fewest number of distinct splitters. Finally, we define and analyze the two-player Splitting Game where one player attempts to construct a splitter while the other player aims to prevent it