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Matematicheskie Voprosy Kriptografii [Mathematical Aspects of Cryptography], 2011, Volume 2, Issue 4, Pages 109–146
DOI: https://doi.org/10.4213/mvk46
(Mi mvk46)
 

Satisfiability of random systems of equations with nonuniform sampling of two-valued unknowns

A. V. Shapovalov

TVP Laboratory, Moscow
References:
Abstract: A random system of $M=M(n)$ equations with $n$ two-valued unknowns is considered. Each equation contains at most $m$ unknowns which are selected randomly. We find conditions on the structure of equations ensuring that for $M=cn^{1-1/r}+o(n^{1-1/r})$, $n\to\infty$, $2\le r\le m+1$, $m=\mathrm{const}$, the probability of existence of solution decreases from 1 to 0 when $c$ increases from 0 to $\infty$. An algorithm detecting the unsolvability of a random system of equations is constructed. This algorithm has low time complexity $O(n^{1-1/r})$, $2\le r\le m+1$. The probability of detecting the unsolvability is asymptotically the same as for the exhaustive search algorithm.
Key words: system of discrete equations, satisfiability, probabilistic algorithms, random hypergraphs.
Received 01.XI.2010
Document Type: Article
UDC: 519.212.2
Language: Russian
Citation: A. V. Shapovalov, “Satisfiability of random systems of equations with nonuniform sampling of two-valued unknowns”, Mat. Vopr. Kriptogr., 2:4 (2011), 109–146
Citation in format AMSBIB
\Bibitem{Sha11}
\by A.~V.~Shapovalov
\paper Satisfiability of random systems of equations with nonuniform sampling of two-valued unknowns
\jour Mat. Vopr. Kriptogr.
\yr 2011
\vol 2
\issue 4
\pages 109--146
\mathnet{http://mi.mathnet.ru/mvk46}
\crossref{https://doi.org/10.4213/mvk46}
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