TR06-043 Authors: Eran Ofek, Uriel Feige

Publication: 26th March 2006 19:03

Downloads: 1826

Keywords:

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.