Recognizing more unsatisfiable random k-sat instances efficiently

Joel Friedman, Andreas Goerdt, Michael Krivelevich

Research output: Contribution to journalArticlepeer-review


It is known that random k-SAT instances with at least cn clauses, where c = ck 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 nk-1/(log n)k-3 clauses can be certified unsatisfiable in polynomial time. We employ spectral methods to improve on this bound. For even k ≥ 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 n3/2+ε clauses, where ε > 0 is an arbitrary constant.

Original languageEnglish
Pages (from-to)408-430
Number of pages23
JournalSIAM Journal on Computing
Issue number2
StatePublished - 2006


  • Random satisfiability
  • Random structures
  • Spectral methods


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