Let @math 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 @math is: first, translate it into a graph @math using a generic reduction from 3-SAT to max-IS, then bound the maximum independent set of @math using the Lovasz @math function. If the @math function returns a value @math , this is a certificate for the unsatisfiability of @math . We show that for random formulas with @math clauses, the above approach fails, i.e. the @math function is likely to return a value of m.
The algorithm considered in the current paper for refuting @math by computing @math was presented in @cite_1 . There is was shown that when @math , almost surely @math . Our current work overcomes a difficulty that prevented the approach of @cite_1 to show that @math , not even for formulas @math with a linear number of clauses. The difficulty was the existence of pairs of clauses that share two variables.
@cite_1
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Let @math be a @math -ary predicate over a finite alphabet. Consider a random CSP @math instance @math over @math variables with @math constraints. When @math the instance @math will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When @math is the @math -ary OR predicate, this is the well studied problem of refuting random @math -SAT formulas, and an efficient algorithm is known only when @math . Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a @math -ary predicate, having @math constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if @math fails to support a @math -wise uniform distribution, then there is an efficient algorithm that refutes random CSP @math instances @math whp when @math . Indeed, our algorithm will "somewhat strongly" refute @math , certifying @math , if @math then we get the strongest possible refutation, certifying @math . This last result is new even in the context of random @math -SAT. Regarding the optimality of our @math requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP @math may be impossible when @math . Thus there is an indication our algorithm's dependence on @math is optimal for every @math , at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the @math -round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz.