### Abstract

A Boolean formula in a conjunctive normal form is called a (k, s) - formula if every clause contains exactly k variables and every variable occurs in at most s clauses. The (k,s) -- SAT problem is the SATISFIABILITY problem restricted to (k,s) -- formulas. It is proved that for every k ≥ 3 there is an integer f(k) such that (k, s) -- SAT is trivial for s ≤ f(k) (because every (k, s) -- formula is satisfiable) and is NP-complete for s ≥ f(k) + 1. Moreover, f(k) grows exponentially with k, namely, [2^{k}/ck] ≤ f(k) ≤ 2^{k-1} - 2^{k-4} - 1 for k ≥ 4.

Original language | English |
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Pages (from-to) | 203-210 |

Number of pages | 8 |

Journal | SIAM Journal on Computing |

Volume | 22 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 1993 |

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

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## Cite this

Kratochvil, J., Savicky, P., & Tuza, Z. (1993). One more occurrence of variables makes satisfiability jump from trivial to NP-complete.

*SIAM Journal on Computing*,*22*(1), 203-210. https://doi.org/10.1137/0222015