ECCC-Report TR06-043https://eccc.weizmann.ac.il/report/2006/043Comments and Revisions published for TR06-043en-usSun, 26 Mar 2006 19:03:24 +0200
Paper TR06-043
| Random 3CNF formulas elude the Lovasz theta function |
Eran Ofek,
Uriel Feige
https://eccc.weizmann.ac.il/report/2006/043Let $\phi$ 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.
Sun, 26 Mar 2006 19:03:24 +0200https://eccc.weizmann.ac.il/report/2006/043