Testing Intersecting and Union-Closed Families

Inspired by the classic problem of Boolean function monotonicity testing, we investigate the testability of other well-studied properties of combinatorial finite set systems, specifically \emph{intersecting} families and \emph{union-closed} families. A function $f: {0,1}ⁿ \to {0,1}$ is intersecting (respectively, union-closed) if its set of satisfying assignments corresponds to an intersecting family (respectively, a union-closed family) of subsets of $[n]$. Our main results are that -- in sharp contrast with the property of being a monotone set system -- the property of being an intersecting set system, and the property of being a union-closed set system, both turn out to be information-theoretically difficult to test. We show that: $\bullet$ For $\epsilon \geq \Omega(1/\sqrt{n})$, any non-adaptive two-sided $\epsilon$-tester for intersectingness must make $2^{{\Omega(n{1/4}/\sqrt{\epsilon})}$} queries. We also give a $2^{{\Omega(\sqrt{n} \log(1/\epsilon)})}$-query lower bound for non-adaptive one-sided $\epsilon$-testers for intersectingness. $\bullet$ For $\epsilon \geq 1/2^{{\Omega(n{0.49})}$,} any non-adaptive two-sided $\epsilon$-tester for union-closedness must make $n^{{\Omega(\log(1/\epsilon))}$} queries. Thus, neither intersectingness nor union-closedness shares the $\mathrm{poly}(n,1/\epsilon)$-query non-adaptive testability that is enjoyed by monotonicity. To complement our lower bounds, we also give a simple $\mathrm{poly}(n^{{\sqrt{n\log(1/\epsilon)}},1/\epsilon)$-query,} one-sided, non-adaptive algorithm for $\epsilon$-testing each of these properties (intersectingness and union-closedness). We thus achieve nearly tight upper and lower bounds for two-sided testing of intersectingness when $\epsilon = \Theta(1/\sqrt{n})$, and for one-sided testing of intersectingness when $\epsilon=\Theta(1).$