The set splittability problem

TL;DR

This paper investigates the set splittability problem, its generalization to arbitrary fractions, and establishes computational complexity results, characterizations for small collections, and the typicality of splittability.

Contribution

It introduces the $p$-splittability problem, proves NP-completeness, provides criteria and complete characterizations for small set collections, and analyzes the prevalence of splittability.

Findings

The $p$-splittability problem is NP-complete.

Complete characterization for three or fewer sets for any $p$.

Splittability is asymptotically prevalent over unsplittability.

Abstract

The set splittability problem is the following: given a finite collection of finite sets, does there exits a single set that contains exactly 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 $\leq1$. We introduce a natural generalization of splittability problem called the $p$-splittability problem, where we replace the fraction $\frac12$ in the definition with an arbitrary fraction $p\in(0,1)$. We first show that the $p$-splittability problem is NP-complete. We then give several criteria for $p$-splittability, including a complete characterization of $p$-splittability for three or fewer sets ($p$ arbitrary), and for four or fewer sets ($p=\frac12$). Finally we prove the asymptotic prevalence of splittability over unsplittability in an appropriate sense.