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Any monotone property of 3-uniform hypergraphs is weakly evasive
Authors
A. Chakrabarti
A.C.-C. Yao
+13 more
C.B. Haselgrove
E. Kushilevitz
F.H. Lutz
H. Buhrman
J. Kahn
N. Linial
N. Nisan
N. Nisan
R. Kulkarni
R. Oliver
R.L. Rivest
T.P. Hayes
Z. Zhang
Publication date
1 January 2013
Publisher
'Springer Science and Business Media LLC'
Doi
Cite
Abstract
For a Boolean function f, let D(f) denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine f. In a classic paper, Rivest and Vuillemin [18] show that any non-constant monotone property P: {0,1} (2n) → {0,1} of n-vertex graphs has D(P) = Ω (n). We extend their result to 3-uniform hypergraphs. In particular, we show that any non-constant monotone property P: {0,1} (3n) → {0,1} of n-vertex 3-uniform hypergraphs has D(P) = Ω (n). Our proof combines the combinatorial approach of Rivest and Vuillemin with the topological approach of Kahn, Saks, and Sturtevant. Interestingly, our proof makes use of Vinogradov's Theorem (weak Gold-bach Conjecture), inspired by its recent use by Babai et. al. [1] in the context of the topological approach. Our work leaves the generalization to k-uniform hypergraphs as an intriguing open question. © Springer-Verlag Berlin Heidelberg 2013
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Last time updated on 01/04/2019
OPUS - University of Technology Sydney
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Last time updated on 13/02/2017