Random 3CNF formulas elude the Lovasz theta function

Uriel Feige, Eran Ofek

Abstract: 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.