### 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 |
---|---|

Pages (from-to) | 203-210 |

Number of pages | 8 |

Journal | SIAM Journal on Computing |

Volume | 22 |

Issue number | 1 |

Publication status | Published - Feb 1993 |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics
- Theoretical Computer Science

### Cite this

*SIAM Journal on Computing*,

*22*(1), 203-210.

**One more occurrence of variables makes satisfiability jump from trivial to NP-complete.** / Kratochvil, Jan; Savicky, Petr; Tuza, Z.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 22, no. 1, pp. 203-210.

}

TY - JOUR

T1 - One more occurrence of variables makes satisfiability jump from trivial to NP-complete

AU - Kratochvil, Jan

AU - Savicky, Petr

AU - Tuza, Z.

PY - 1993/2

Y1 - 1993/2

N2 - 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, [2k/ck] ≤ f(k) ≤ 2k-1 - 2k-4 - 1 for k ≥ 4.

AB - 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, [2k/ck] ≤ f(k) ≤ 2k-1 - 2k-4 - 1 for k ≥ 4.

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UR - http://www.scopus.com/inward/citedby.url?scp=0027542444&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0027542444

VL - 22

SP - 203

EP - 210

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 1

ER -