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A Linear Characterization of NP-Complete Problems

  • Conference paper
7th International Conference on Automated Deduction (CADE 1984)

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

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Abstract

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.

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References

  • H. C. Andrews, J. Kane, Kronecker matrices, computer implementation, and generalized spectra, Journal of the ACM 17 (1970) 260–268.

    Article  MathSciNet  MATH  Google Scholar 

  • R. F. Arnold, M. A. Harrison, Algebraic properties of symmetric and partially symmetric boolean functions, IEEE Transactions on Electronic Computers 12 (1963) 244–251.

    Article  MathSciNet  MATH  Google Scholar 

  • M. L. Balinski, Integer programming: methods, uses, computation, Management Science 12 (1965) 253–313.

    Article  MathSciNet  MATH  Google Scholar 

  • K. G. Beauchamp, Walsh functions and their applications, Academic Press, London, 1975.

    MATH  Google Scholar 

  • K. Bing, On simplifying truth-functional formulas, Journal of Symbolic Logic 21, (1956) 253–254.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • F. M. Brown, The origin of the method of iterated consensus, IEEE Transactions on Electronic Computers 17 (1968) 802.

    Article  Google Scholar 

  • J. R. Buchi, Turing-machines and the Entscheidungsproblem, Math. Annalen 148 (1962) 201–213.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Charnes, Optimality and degeneracy in linear programming, Econometrica 20 (1962) 160–170.

    Article  MathSciNet  MATH  Google Scholar 

  • S. A. Cook, The complexity of theorem-proving procedures, Third ACM Symposium on Theory of Computing, (1971) 151–158.

    Google Scholar 

  • C. H. Cunkle, Symmetric boolean functions, American Math. Montly 70 (1963) 833–836.

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Galil, On the complexity of regular resolution and the Davis-Putnam procedure, Theoretical Computer Science 4 (1977) 23–46.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MATH  Google Scholar 

  • 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 

  • 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 

  • M. Grötschel, Approaches to hard combinatorial optimization problems, in: B. Korte, ed., Modern Appl. Math., Optimization and Operations Research (1982) 437–515.

    Google Scholar 

  • M. Grötschel, L. Lovasz, A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatoritca 1 (1981) 169–197.

    Article  MathSciNet  MATH  Google Scholar 

  • R. M. Karp, C. H. Papadimitriou, On linear characterizations of combinatorial optimization problems, SIAM J. Computing 11 (1982) 620–632.

    Article  MathSciNet  MATH  Google Scholar 

  • R. E. Gomory, The traveling salesman problem, in: Proc. IBM Scientific Computing Symposium on Combinatorial Problems (1964) 93–121.

    Google Scholar 

  • R. M. Karp, Reducibility among combinatorial problems, in: R. E. Miller, J, W, Thacher, eds., Complexity of Copmputer Computations (1972) 85–103.

    Google Scholar 

  • L. G. Khachian, A polynomial algorithm in linear programming, Soviet Mathematics Doklady 20 (1979) 191–194.

    Google Scholar 

  • D. E. Knuth, The art of computer programming, volume 1, Addison-Wesley 1968.

    Google Scholar 

  • H. W. Lenstra, Integer programming with a fixed number of variables, Report 81-03, University of Amsterdam, 1981.

    Google Scholar 

  • L. Löwenheim, Uber das Auflosungsproblem im logischen Klassenkalkul, Sitzungsberichte der Berliner Mathematischen Gesellschaft 7 (1908) 89–94.

    MATH  Google Scholar 

  • M. W. Padberg, ed., Combinatorial optimization, in; Math. Programming Study 12, North-Holland 1980, pp 1–221.

    Google Scholar 

  • C. H. Papadimitriou, On the complexity of integer programming, Journal of the ACM 28 (1981) 765–768.

    Article  MathSciNet  MATH  Google Scholar 

  • W. R. Pulleyblank, Polyhedral combinatorics, in: Mathematical Programming, the State of the Art, Springer-Verlag 1983, pp 312–345.

    Google Scholar 

  • J. Riordan, Combinatorial Identities, John Wiley and Sons 1968.

    Google Scholar 

  • W. V. Quine, Away to simplify truth functions, Am. Math. Montly 62 (1955) 627–631.

    Article  MATH  Google Scholar 

  • J. A. Robinsion, A machine-oriented logic based on the resolution principle, Journal of the ACM 12 (1965) 23–41.

    Article  MathSciNet  Google Scholar 

  • W. Semon, E-algebras in switching theory, Transactions AIEE 80 (1961) 265–269.

    Google Scholar 

  • C. E. Shannon, A symbolic analysis of relay and switching circuits, Transactions AIEE 57 (1938) 713–723.

    Google Scholar 

  • T. Skolem, Uber die symmetrisch allgemeinen Losungen im Klassenkalkul, Fundamenta Mathematicae 18 (1932) 61–76.

    Article  MATH  Google Scholar 

  • D. Slepian, On the number of symmetry types of boolean functions of n variables, Canadian J, of Math. 5 (1953) 185–193.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Stoer, C. Witzgall, Convexity and Optimization in Finite Dimensions I, Springer-Verlag 1970.

    Google Scholar 

  • 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 

  • 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 

  • S. Ursic, A discrete optimization algorithm nased on binomial polytopes, Ph. D. Thesis, University of Wisconsin, Madison, 1975.

    Google Scholar 

  • 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 

  • S. Ursic, The ellipsoid algorithm for linear inequalities in exact arithmetic, IEEE Foundations of Computer Science 23 (1982) 321–326.

    MathSciNet  Google Scholar 

  • 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 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • R. O. Winder, Symmetry types in threshold logic, IEEE Transactions on Computers 17 (1968) 75–78.

    Article  MATH  Google Scholar 

  • M. C. Pease III, Methods of matrix algebra, Academic Press 1965.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Madison, Wisconsin

    Silvio Ursic

Authors
  1. Silvio Ursic
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Editor information

Editors and Affiliations

  1. SRI International, 333 Ravenswood Avenue, Menlo Park, CA, 94025, USA

    R. E. Shostak

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© 1984 Springer-Verlag Berlin Heidelberg

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Ursic, S. (1984). A Linear Characterization of NP-Complete Problems. In: Shostak, R.E. (eds) 7th International Conference on Automated Deduction. CADE 1984. Lecture Notes in Computer Science, vol 170. Springer, New York, NY. https://doi.org/10.1007/978-0-387-34768-4_5

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  • DOI: https://doi.org/10.1007/978-0-387-34768-4_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-96022-7

  • Online ISBN: 978-0-387-34768-4

  • eBook Packages: Springer Book Archive

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