The set splittability problem

Abstract

The set splittability problem is the following: given a finite collection of finite sets, does there exits a single set that selects half the elements from each set in the collection? (If a set has odd size, we allow the floor or ceiling.) It is natural to study the set splittability problem in the context of combinatorial discrepancy theory and its applications, since a collection is splittable if and only if it has discrepancy ≤ 1.We introduce a variant of the splittability problem called the p-splittability problem, in which one seeks to select the fraction p from each set instead of half. We show that the psplittability problem is NP-complete. We then investigate several criteria for p-splittability, giving a complete characterization of p-splittability for three or fewer sets. In the case of ordinary set splittability (p = 1 2 ) we extend the characterization to four or fewer sets. Finally we show that when there are sufficiently many elements, unsplittability is asymptotically much more rare than splittability.