Foundations and Trends® in Theoretical Computer Science
An efficient sparse regularity concept
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Complexity of propositional proofs under a promise
ACM Transactions on Computational Logic (TOCL)
Public-key cryptography from different assumptions
Proceedings of the forty-second ACM symposium on Theory of computing
An Efficient Sparse Regularity Concept
SIAM Journal on Discrete Mathematics
Towards non-black-box lower bounds in cryptography
TCC'11 Proceedings of the 8th conference on Theory of cryptography
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Short Propositional Refutations for Dense Random 3CNF Formulas
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
Complexity of propositional proofs under a promise
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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We consider random 3CNF formulas with n variables and m clauses. It is well known that when m \ge cn (for a sufficiently large constant c), most formulas are not satisfiable. However, it is not known whether such formulas are likely to have polynomial size witnesses that certify that they are not satisfiable. A value of m\simeqn^{3/2} was the forefront of our knowledge in this respect. When m \ge cn^{3/2}, such witnesses are known to exist, based on spectral techniques. When m \le n^{3/2- \in}, it is known that resolution (which is a common approach for refutation) cannot produce witnesses of size smaller than 2^{n^ \in}. Likewise, it is known that certain variants of the spectral techniques do not work in this range. In the current paper we show that when m \ge cn^{7/5}, almost all 3CNF formulas have polynomial size witnesses for non-satisfiability. We also show that such a witness can be found in time 2^{O\left( {n^{0.2} \log n} \right)} , whenever it exists. Our approach is based on an extension of the known spectral techniques, and involves analyzing a certain fractional packing problem for random 3-uniform hypergraphs.