A Linear Characterization of NP-Complete Problems

  • Silvio Ursic
Part of the Lecture Notes in Computer Science book series (LNCS, volume 170)


We present a linear characterization for the solution sets of propositional calculus formulas in conjunctive normal form. We obtain recursive definitions for the linear characterization similar to the basic recurrence relation used to define binomial coefficients. As a consequence, we are able to use standard combinatorial and linear algebra techniques to describe properties of the linear characterization.


Boolean Function Turing Machine Conjunctive Normal Form Truth Assignment Boolean Formula 
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  1. H. C. Andrews, J. Kane, Kronecker matrices, computer implementation, and generalized spectra, Journal of the ACM 17 (1970) 260–268.MathSciNetCrossRefzbMATHGoogle Scholar
  2. R. F. Arnold, M. A. Harrison, Algebraic properties of symmetric and partially symmetric boolean functions, IEEE Transactions on Electronic Computers 12 (1963) 244–251.MathSciNetCrossRefzbMATHGoogle Scholar
  3. M. L. Balinski, Integer programming: methods, uses, computation, Management Science 12 (1965) 253–313.MathSciNetCrossRefzbMATHGoogle Scholar
  4. K. G. Beauchamp, Walsh functions and their applications, Academic Press, London, 1975.zbMATHGoogle Scholar
  5. K. Bing, On simplifying truth-functional formulas, Journal of Symbolic Logic 21, (1956) 253–254.MathSciNetCrossRefzbMATHGoogle Scholar
  6. A. Blake, Canonical expressions in boolean algebra, Ph. D. thesis, Dept. of Mathematics, University of Chicago, August 1937. Review in: J. Symb. Logic 3 (1938) 93.Google Scholar
  7. A. Blake, A boolean derivation of the Moore-Osgood theorem, Journal of Symbolic Logic 11 (1946) 65–70. Review in: J. Symb. Logic 12 (1947) 89–90.MathSciNetCrossRefzbMATHGoogle Scholar
  8. F. M. Brown, The origin of the method of iterated consensus, IEEE Transactions on Electronic Computers 17 (1968) 802.CrossRefGoogle Scholar
  9. J. R. Buchi, Turing-machines and the Entscheidungsproblem, Math. Annalen 148 (1962) 201–213.MathSciNetCrossRefzbMATHGoogle Scholar
  10. A. Charnes, Optimality and degeneracy in linear programming, Econometrica 20 (1962) 160–170.MathSciNetCrossRefzbMATHGoogle Scholar
  11. S. A. Cook, The complexity of theorem-proving procedures, Third ACM Symposium on Theory of Computing, (1971) 151–158.Google Scholar
  12. C. H. Cunkle, Symmetric boolean functions, American Math. Montly 70 (1963) 833–836.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Z. Galil, On the complexity of regular resolution and the Davis-Putnam procedure, Theoretical Computer Science 4 (1977) 23–46.MathSciNetCrossRefzbMATHGoogle Scholar
  14. M. R. Garey, D. S. Johnson, Computers and intractability, a guide to the theory of NP-completeness, W. H. Freeman and company, San Francisco, 1979.zbMATHGoogle Scholar
  15. I. J. Good, The interaction algorithm and practical Fourier analysis, J. Royal Stat. Soc. Ser. B (1958) 361–372, ibid. (1960) 372-375.Google Scholar
  16. E. Goto, H. Takahasi, Some theorems useful in threshold logic for enumerating boolean functions, Proc. IFIP Congress 62, North-Holland, Amsterdam 1963, pp 747–752.Google Scholar
  17. M. Grötschel, Approaches to hard combinatorial optimization problems, in: B. Korte, ed., Modern Appl. Math., Optimization and Operations Research (1982) 437–515.Google Scholar
  18. M. Grötschel, L. Lovasz, A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatoritca 1 (1981) 169–197.MathSciNetCrossRefzbMATHGoogle Scholar
  19. R. M. Karp, C. H. Papadimitriou, On linear characterizations of combinatorial optimization problems, SIAM J. Computing 11 (1982) 620–632.MathSciNetCrossRefzbMATHGoogle Scholar
  20. R. E. Gomory, The traveling salesman problem, in: Proc. IBM Scientific Computing Symposium on Combinatorial Problems (1964) 93–121.Google Scholar
  21. R. M. Karp, Reducibility among combinatorial problems, in: R. E. Miller, J, W, Thacher, eds., Complexity of Copmputer Computations (1972) 85–103.Google Scholar
  22. L. G. Khachian, A polynomial algorithm in linear programming, Soviet Mathematics Doklady 20 (1979) 191–194.Google Scholar
  23. D. E. Knuth, The art of computer programming, volume 1, Addison-Wesley 1968.Google Scholar
  24. H. W. Lenstra, Integer programming with a fixed number of variables, Report 81-03, University of Amsterdam, 1981.Google Scholar
  25. L. Löwenheim, Uber das Auflosungsproblem im logischen Klassenkalkul, Sitzungsberichte der Berliner Mathematischen Gesellschaft 7 (1908) 89–94.zbMATHGoogle Scholar
  26. M. W. Padberg, ed., Combinatorial optimization, in; Math. Programming Study 12, North-Holland 1980, pp 1–221.Google Scholar
  27. C. H. Papadimitriou, On the complexity of integer programming, Journal of the ACM 28 (1981) 765–768.MathSciNetCrossRefzbMATHGoogle Scholar
  28. W. R. Pulleyblank, Polyhedral combinatorics, in: Mathematical Programming, the State of the Art, Springer-Verlag 1983, pp 312–345.Google Scholar
  29. J. Riordan, Combinatorial Identities, John Wiley and Sons 1968.Google Scholar
  30. W. V. Quine, Away to simplify truth functions, Am. Math. Montly 62 (1955) 627–631.CrossRefzbMATHGoogle Scholar
  31. J. A. Robinsion, A machine-oriented logic based on the resolution principle, Journal of the ACM 12 (1965) 23–41.MathSciNetCrossRefGoogle Scholar
  32. W. Semon, E-algebras in switching theory, Transactions AIEE 80 (1961) 265–269.Google Scholar
  33. C. E. Shannon, A symbolic analysis of relay and switching circuits, Transactions AIEE 57 (1938) 713–723.Google Scholar
  34. T. Skolem, Uber die symmetrisch allgemeinen Losungen im Klassenkalkul, Fundamenta Mathematicae 18 (1932) 61–76.CrossRefzbMATHGoogle Scholar
  35. D. Slepian, On the number of symmetry types of boolean functions of n variables, Canadian J, of Math. 5 (1953) 185–193.MathSciNetCrossRefzbMATHGoogle Scholar
  36. J. Stoer, C. Witzgall, Convexity and Optimization in Finite Dimensions I, Springer-Verlag 1970.Google Scholar
  37. J. J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign-successions, and tesselated pavements in two or more colours, with applications to Newton’s rule, ornamental tile-work, and the theory of numbers, Philisophical Magazine 34 (1867) 461–475. (Cormment; part II, which the author says will follow, seems to have never been published)Google Scholar
  38. G. S. Tseitin, On the complexity of derivations in propositional calculus, Studies in constructive mathematics and mathematical logic, ed. A. O. Slisenko, Vol. 8 (1968) 115–125.Google Scholar
  39. S. Ursic, A discrete optimization algorithm nased on binomial polytopes, Ph. D. Thesis, University of Wisconsin, Madison, 1975.Google Scholar
  40. S. Ursic, Binomial polytopes and NP-complete problems, in J, F, Traub, ed., Algorithms and Complexity, New directions and Recent Results, Academic Press 1976.Google Scholar
  41. S. Ursic, The ellipsoid algorithm for linear inequalities in exact arithmetic, IEEE Foundations of Computer Science 23 (1982) 321–326.MathSciNetGoogle Scholar
  42. S. Ursic, A linear characterization of NP-complete problems, Proceedings of the twenty-first annual Allerton conference on communication, control, and computing, (1983) 100–109.Google Scholar
  43. A. N. Whitehead, Memoir on the algebra of symbolic logic, American Journal of Math. 23 (1901) 139–165. Ibid. Part II 23 (1901) 297-316.MathSciNetCrossRefzbMATHGoogle Scholar
  44. R. O. Winder, Symmetry types in threshold logic, IEEE Transactions on Computers 17 (1968) 75–78.CrossRefzbMATHGoogle Scholar
  45. M. C. Pease III, Methods of matrix algebra, Academic Press 1965.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Silvio Ursic
    • 1
  1. 1.Madison

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