A complexity index for satisfiability problems

• Published in: SIAM journal on computing Vol. 23; no. 1; pp. 45 - 49
• Main Authors: BOROS, E, CRAMA, Y, HAMMER, P. L, SAKS, M
• Format: Journal Article Web Resource
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.02.1994; Society for Industrial & Applied Mathematics
• Subjects: Algorithms; Applied sciences; Artificial intelligence; Boolean; Boolean satisfiability; Computer science; Computer science; control theory; systems; Engineering, computing & technology; Exact sciences and technology; Grants; Ingénierie, informatique & technologie; Linear programming; Mathematical programming; Mathematics; Mathématiques; Miscellaneous; NP completeness; Operational research and scientific management; Operational research. Management science; Physical, chemical, mathematical & earth Sciences; Physique, chimie, mathématiques & sciences de la terre; polynomial algorithms; Sciences informatiques; Variables; NP complete problem; Boolean programming; Mathematical programming
• ISSN: 0097-5397; 1095-7111; 1095-7111
• DOI: 10.1137/S0097539792228629

This paper associates a linear programming problem (LP) to any conjunctive normal form $\phi $, and shows that the optimum value $Z(\phi )$ of this LP measures the complexity of the corresponding ${\textit{SAT}}$ (Boolean satisfiability) problem. More precisely, there is an algorithm for ${\textit{SAT}}$ that runs in polynomial time on the class of satisfiability problems satisfying $Z(\phi ) \leqslant 1 + \tfrac{{c\log n}}{n}$ for a fixed constant $c$, where $c$ is the number of variables. In contrast, for any fixed $\beta < 1$, $SAT$ is still NP complete when restricted to the class of CNFs for which $Z(\phi ) \leqslant 1 + ({1 / {n^\beta }})$.