A New Lower Bound of Critical Function for (k,s)-SAT

  • Ping Gong
  • Daoyun Xu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3959)


(k,s)–SAT is the propositional satisfiable problem restricted to the instance where each clause has exactly k distinct literals and every variable occurs at most s times. It is known that there exits a critical function f such that for sf(k), all (k,s)–SAT instances are satisfiable, but (k,f(k)+1)–SAT is already NP–complete(k≥ 3). It’s open whether f is computable. In this paper, analogous to the randomized algorithm for finding a two-coloring for given uniform k–hypergraph, the similar one for outputting an assignment for a given formula is presented. Based on it and the probabilistic method, we prove, for every integer k≥ 2, each formula F in (k, *)–CNF with less than 0.58 \(\times \sqrt{\frac{k}{{\rm ln} k}}2^k\) clauses is satisfiable. In addition, by the Lovász Local Lemma, we improve the previous result about the lower bound of f(k).


Probabilistic Method Random Assignment Critical Function Conjunctive Normal Form Propositional Variable 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ping Gong
    • 1
  • Daoyun Xu
    • 2
  1. 1.Department of MathematicsGuizhou UniversityGuiyangP.R. China
  2. 2.Department of Computer ScienceGuizhou UniversityGuiyangP.R. China

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