An efficient sparse regularity concept
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
An Efficient Sparse Regularity Concept
SIAM Journal on Discrete Mathematics
Short Propositional Refutations for Dense Random 3CNF Formulas
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
Approximating independent set in perturbed graphs
Discrete Applied Mathematics
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It is known that random $k$-SAT instances with at least $cn$ clauses, where $c =\nobreak c_k$ is a suitable constant, are unsatisfiable (with high probability). We consider the problem to certify efficiently the unsatisfiability of such formulas. A backtracking-based algorithm of Beame et al. [SIAM J. Comput.,} 31 (2002), pp. 1048--1075] shows that $k$-SAT instances with at least $n^{k-1}/(\log n)^{k-2}$ clauses can be certified unsatisfiable in polynomial time. We employ spectral methods to improve on this bound. For even $k\ge 4$ we present a polynomial time algorithm which certifies random $k$-SAT instances with at least $n^{(k/2)+o(1)}$ clauses as unsatisfiable (with high probability). For odd $k$ we focus on 3-SAT instances and obtain an efficient algorithm for formulas with at least $n^{3/2+\varepsilon}$ clauses, where $\varepsilon 0$ is an arbitrary constant.