TR03-030 Authors: Amin Coja-Oghlan, Andreas Goerdt, AndrÃ© Lanka, Frank SchÃ¤dlich

Publication: 23rd April 2003 21:27

Downloads: 2233

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Abstract. It is known that random k-SAT formulas with at least

(2^k*ln2)*n random clauses are unsatisfiable with high probability. This

result is simply obtained by bounding the expected number of satisfy-

ing assignments of a random k-SAT instance by an expression tending

to 0 when n, the number of variables tends to infinity. This argument

does not give us an efficient algorithm certifying the unsatisfiability of a

given random instance. For even k it is known that random k-SAT in-

stances with at least Poly(logn)*n^(k/2) clauses can be efficiently certified

as unsatisfiable. For k = 3 we need at least n^((3/2)+epsilon) random clauses.

In case of even k we improve the aforementioned results in two ways.

There exists a constant C such that 4-SAT instances with at least C*n^2

clauses can be efficiently certified as unsatisfiable. Moreover, we give

a satisfiability algorithm which runs in expected polynomial time over

all k-SAT instances with C*n^(k/2) clauses. Our proofs are based on the

direct application of known approximation algorithms on the one hand,

and on a recent estimate of the theta-function for random graphs with a

linear number of edges, on the other hand.