A simple first moment argument shows that in a randomly chosen k-SAT formula with m clauses over n boolean variables, the fraction of satisfiable clauses is at most 1–2 − − k+o(1) as m/n→∞ almost surely. In this paper, we deal with the corresponding algorithmic strong refutation problem: given a random k-SAT formula, can we find a certificate that the fraction of satisfiable clauses is at most 1–2 − −  k+o(1) in polynomial time? We present heuristics based on spectral techniques that in the case k=3, m≥ln (n)6 n 3/2 and in the case k=4, mCn 2 find such certificates almost surely, where C denotes a constant. Our methods also apply to some hypergraph problems.


Polynomial Time Algorithm Chromatic Number Triple System Satisfying Assignment Spectral Technique 
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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Andreas Goerdt
    • 2
  • André Lanka
    • 2
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Fakultät für InformatikTechnische Universität ChemnitzChemnitzGermany

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