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TR06-043 | 22nd March 2006 00:00

Random 3CNF formulas elude the Lovasz theta function


Authors: Eran Ofek, Uriel Feige
Publication: 26th March 2006 19:03
Downloads: 2840


Let $\phi$ be a 3CNF formula with n variables and m clauses. A
simple nonconstructive argument shows that when m is
sufficiently large compared to n, most 3CNF formulas are not
satisfiable. It is an open question whether there is an efficient
refutation algorithm that for most such formulas proves that they
are not satisfiable. A possible approach to refute a formula
$\phi$ is: first, translate it into a graph $G_{\phi}$ using a
generic reduction from 3-SAT to max-IS, then bound the maximum
independent set of $G_{\phi}$ using the Lovasz $\vartheta$
function. If the $\vartheta$ function returns a value < m, this
is a certificate for the unsatisfiability of $\phi$. We show that
for random formulas with $m < n^{3/2 -o(1)}$ clauses, the above
approach fails, i.e. the $\vartheta$ function is likely to return
a value of m.

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