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 Prikl. Diskr. Mat., 2020, Number 48, Pages 16–21 (Mi pdm701)

Theoretical Backgrounds of Applied Discrete Mathematics

One-to-one correspondense between proper families of Boolean functions and unique sink orientations of cubes

K. D. Tsaregorodtsev

Moscow State University, Moscow, Russia

Abstract: In the paper, we study the relationship between proper families of Boolean functions and unique sink orientations of Boolean cubes. A family of Boolean functions $F = (f_1(x_1, \ldots, x_n), \ldots, f_n(x_1, \ldots, x_n))$ is called proper if for every two binary vectors $\alpha, \beta$, $\alpha \ne \beta$, the following condition holds:
$$\exists i (\alpha_i \ne \beta_i & f_i(\alpha) = f_i(\beta)).$$
Unique sink orientation of Boolean cube $\mathbb{E}_n$ is such an orientation of edges of $\mathbb{E}_n$ that any subcube of $\mathbb{E}_n$ has a unique sink, i.e., a unique vertex without outgoing edges. The existence of one-to-one correspondence between two classes of objects is proved, and various properties are derived for proper families. The following boundary for the number $T(n)$ of proper families of given size $n$ is obtained: there exist two numbers $B$ and $A$, $B \ge A > 0$, such that $n^{A 2^n} \le T(n) \le n^{B 2^n}$ for $n \ge 2$. Also, coNP-completeness of the problem of recognizing properness is derived.

Keywords: proper families of Boolean functions, unique sink orientations.

DOI: https://doi.org/10.17223/20710410/48/2

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UDC: 519.1+512.5

Citation: K. D. Tsaregorodtsev, “One-to-one correspondense between proper families of Boolean functions and unique sink orientations of cubes”, Prikl. Diskr. Mat., 2020, no. 48, 16–21

Citation in format AMSBIB
\Bibitem{Tsa20} \by K.~D.~Tsaregorodtsev \paper One-to-one correspondense between proper families of~Boolean functions and unique sink orientations of~cubes \jour Prikl. Diskr. Mat. \yr 2020 \issue 48 \pages 16--21 \mathnet{http://mi.mathnet.ru/pdm701} \crossref{https://doi.org/10.17223/20710410/48/2} 

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