On the Sensitivity Complexity of $k$-Uniform Hypergraph Properties

Abstract

In this paper we investigate the sensitivity complexity of hypergraph properties. We present a $k$-uniform hypergraph property with sensitivity complexity $O(n^{\lceil k/3\rceil })$ for any $k\ge 3$, where $n$ is the number of vertices. Moreover, we can do better when $k\equiv 1$ (mod 3) by presenting a $k$-uniform hypergraph property with sensitivity $O(n^{\lceil k/3\rceil -1/2})$. This result disproves a conjecture of Babai~\cite{Babai}, which conjectures that the sensitivity complexity of $k$-uniform hypergraph properties is at least $\Omega (n^{k/2})$. We also investigate the sensitivity complexity of other symmetric functions and show that for many classes of transitive Boolean functions the minimum achievable sensitivity complexity can be $O(N^{1/3})$, where $N$ is the number of variables. Finally, we give a lower bound for sensitivity of $k$-uniform hypergraph properties, which implies the {\em sensitivity conjecture} of $k$-uniform hypergraph properties for any constant $k$.

References in corpus (2)

• Evasiveness and the Distribution of Prime Numbers
• Lattice Variant of the Sensitivity Conjecture

Cited by in corpus (1)

• Lower bounds for sensitivity of graph properties