Spectra, Euclidean representations and clusterings of hypergraphs
Discrete Mathematics
Approximability of maximum splitting of k-sets and some other Apx-complete problems
Information Processing Letters
Derandomizing Approximation Algorithms Based on Semidefinite Programming
SIAM Journal on Computing
The phase transition in 1-in-k SAT and NAE 3-SAT
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Some optimal inapproximability results
Journal of the ACM (JACM)
Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
MFCS '02 Proceedings of the 27th International Symposium on Mathematical Foundations of Computer Science
Techniques from combinatorial approximation algorithms yield efficient algorithms for random 2k-SAT
Theoretical Computer Science
MAX k-CUT and approximating the chromatic number of random graphs
Random Structures & Algorithms
On the hardness and easiness of random 4-SAT formulas
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
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It is known that random $k$-Sat instances with at least $(2^k \cdot \ln 2)\cdot n$ random clauses are unsatisfiable with high probability. This result is simply obtained by showing that the expected number of satisfying assignments tends to $0$ when the number of variables $n$ tends to infinity. This proof does not directly provide us with an efficient algorithm certifying the unsatisfiability of a given random formula. Concerning efficient algorithms, it is essentially known that random formulas with $n^\varepsilon \cdot n^{k/2}$ clauses with $k$ literals can be efficiently certified as unsatisfiable. The present paper is the result of trying to lower this bound. We obtain better bounds for some specialized satisfiability problems. These results are based on discrepancy investigations for hypergraphs.Further, we show that random formulas with a linear number of clauses can be efficiently certified as unsatisfiable in the Not-All-Equal-$3$-Sat sense. A similar result holds for the non-$3$-colourability of random graphs with a linear number of edges. We obtain these results by direct application of approximation algorithms.