Linear Programming Formulation of Boolean Satisfiability Problem

  • Algirdas Antano Maknickas
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 275)


It was investigated the Boolean satisfiability (SAT) problem defined as follows: given a Boolean formula, check whether an assignment of Boolean values to the propositional variables in the formula exists, such that the formula evaluates to true. If such an assignment exists, the formula is said to be satisfiable; otherwise, it is unsatisfiable. With using of analytical expressions of multi-valued logic 2SAT boolean satisfiability was formulated as linear programming optimization problem. The same linear programming formulation was extended to find 3SAT and kSAT boolean satisfiability for k greater than 3. So, using new analytic multi-valued logic expressions and linear programming formulation of boolean satisfiability proposition that kSAT is in P and could be solved in linear time was proved.


Boolean satisfiability Conjunctive normal form Convex function Linear programming Multi-valued logic NP problem Time complexity. 


  1. 1.
    Adler I, Karmarkar N, Resende MGC, Veiga G (1989) An implementation of Karmarkar’s algorithm for linear programming. Math Program 44:297–335MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aspvall B, Plass MF, Tarjan RE (1979) A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf Proces Lett 8(3):121–123MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cheriyan J, Mehlhorn K (1996) Algorithms for dense graphs and networks on the random access computer. Algorithmica 15(6):521–549MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cook S (1971) The complexity of theorem proving procedures. In: Proceedings of the third annual ACM symposium on theory of, computing, pp 151–158Google Scholar
  5. 5.
    Diaby M (2010) Linear programming formulation of the set partitioning problem. Int J Oper Res 8(4):399–427MathSciNetzbMATHGoogle Scholar
  6. 6.
    Diaby M (2010) Linear programming formulation of the vertex colouring problem. Int J Math Oper Res 2(3):259–289MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Even S, Itai A, Shamir A (1976) On the complexity of time table and multi-commodity flow problems. SIAM J Comput 5(4):691–703MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Krom MR (1967) The decision problem for a class of first-order formulas in which all disjunctions are binary. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 13:15–20MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Levin L (1973) Universal search problems. Probl Inf Trans 9(3), 265–266. (Russian)Google Scholar
  10. 10.
    Maknickas AA (2010) Finding of k in Fagin’s R. Theorem 24. arXiv:1012.5804v1 (2010).Google Scholar
  11. 11.
    Maknickas AA (2012) How to solve kSAT in polinomial time. arXiv:1203.6020v1Google Scholar
  12. 12.
    Maknickas AA (2012) How to solve kSAT in polinomial time. arXiv:1203.6020v2Google Scholar
  13. 13.
    Maknickas AA (2013) Programming formulation of kSAT. In: Lecture notes in engineering and computer science: Proceedings of the international multiConference of engineers and computer scientists 2013, pp. 1066–1070Google Scholar
  14. 14.
    Tarjan RE (1972) Depth-first search and linear graph algorithms. SIAM J Comput 1(2):146–160MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Vilnius Gediminas Technical UniversityVilniusLithuania

Personalised recommendations