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Some Results on Random Unsatisfiable k-Sat Instances and Approximation Algorithms Applied to Random Structures

Published online by Cambridge University Press:  20 May 2003

ANDREAS GOERDT
Affiliation:
Technische Universität Chemnitz, Fakultät für Informatik, Straße der Nationen 62, 09107 Chemnitz, Germany (e-mail: goerdt@informatik.tu-chemnitz.de)
TOMASZ JURDZIŃSKI
Affiliation:
Wroclaw University, Institute of Computer Science, Przesmyckiego 20, 50-151 Wroclaw, Poland (e-mail: tju@ii.uni.wroc.pl)

Abstract

It is known that random $k$-Sat instances with at least $(2^k \cdot \ln 2)\cdot n$ random clauses are unsatisfiable with high probability. This result is simply obtained by showing that the expected number of satisfying assignments tends to $0$ when the number of variables $n$ tends to infinity. This proof does not directly provide us with an efficient algorithm certifying the unsatisfiability of a given random formula. Concerning efficient algorithms, it is essentially known that random formulas with $n^\varepsilon \cdot n^{k/2}$ clauses with $k$ literals can be efficiently certified as unsatisfiable. The present paper is the result of trying to lower this bound. We obtain better bounds for some specialized satisfiability problems. These results are based on discrepancy investigations for hypergraphs.

Further, we show that random formulas with a linear number of clauses can be efficiently certified as unsatisfiable in the Not-All-Equal-$3$-Sat sense. A similar result holds for the non-$3$-colourability of random graphs with a linear number of edges. We obtain these results by direct application of approximation algorithms.

Type
Research Article
Copyright
© 2003 Cambridge University Press

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