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arXiv · 1611.01542

The set splittability problem

Peter Bernstein·Cashous Bortner·Samuel Coskey·Shuni Li·Connor Simpson
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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.

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Peter Bernstein, Cashous Bortner, Samuel Coskey, Shuni Li, Connor Simpson. 2019-08-14. The set splittability problem. https://arxiv.org/abs/1611.01542

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