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On the Sensitivity Complexity of k-Uniform Hypergraph Properties

Authors Qian Li, Xiaoming Sun

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Qian Li
Xiaoming Sun

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Qian Li and Xiaoming Sun. On the Sensitivity Complexity of k-Uniform Hypergraph Properties. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 51:1-51:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.STACS.2017.51

Abstract

In this paper we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n^{ceil(k/3)}) for any k >= 3, where n is the number of vertices. Moreover, we can do better when k = 1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n^{ceil(k/3)-1/2}). This result disproves a conjecture of 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 weakly symmetric functions and show that for many classes of transitive-invariant 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 sensitivity conjecture of k-uniform hypergraph properties for any constant k.
Keywords
  • Sensitivity Complexity
  • k-uniform Hypergraph Properties
  • Boolean Function
  • Turan's question

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