On the Computational Complexity of Integer Programming Problems

  • Ravindran Kannan
  • Clyde L. Monma
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 157)


Recently much effort has been devoted to determining the computational complexity for a variety of integer programming problems. In this paper a general integer programming problem is shown to be NP-complete; the proof given for this result uses only elementary linear algebra. Complexity results are also summarized for several particularizations of this general problem, including knapsack problems, problems which relax integrality or non-negativity restrictions and integral optimization problems with a fixed number of variables.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A.J. Ano, J.E. Hopcroft, J.D. Ullman. The Design and Analysis of Computer Algorithms, Addison Wesley, 1974.Google Scholar
  2. [2]
    A. Bachern, personal communication.Google Scholar
  3. [3]
    I. Borosh, L.B. Treybig, Bounds on Positive Solutions of Linear Diophantine Equations, Proc. Amer. Math. Soc., Vol. 55, 299, 1976.CrossRefGoogle Scholar
  4. [4]
    G.H. Bradley, Algorithm and Bound for the Greatest Common Divisor of n Integer, Comm. ACM 13, 433–436, 1970.CrossRefGoogle Scholar
  5. [5]
    G.H. Bradley, Algorithms for Hermite and Smith Normal Matrices and Linear Diophantine Equations, Mathematics Computation 25, No. 116, 897–907, 1971.CrossRefGoogle Scholar
  6. [6]
    S.A. Cook, The complexity of theorem proving procedures, Proc. Third ACM Symp. on Th. of Computation (1971), 151–158.Google Scholar
  7. [7]
    S.A. Cook, A short proof that the linear diophantine problem is in NP, Oct., 1976, Unpublished.Google Scholar
  8. [8]
    J. Edmonds, F.R. Giles, A Min-Max Relation for Submodular Functions on Graphs, Annals of Discrete Mathematics 1, 185–204, 1977.CrossRefGoogle Scholar
  9. [9]
    M.R. Garey, D.S. Johnson, Computers and Intractability : A. Guide to the Theory of NP-Completeness, to appear W.H. Freeman, publisher, 1978.Google Scholar
  10. [10]
    R.S. Garfinkel, G.L. Nemhauser, Integer Programming, John Wiley, 1972.Google Scholar
  11. [11]
    J. Gathen, M. Sieveking, Linear integer inequalities are NP-Complete, submitted to SIAM J. Computing.Google Scholar
  12. [12]
    P.C. Gilmore, R.E. Gomory, The Theory and Computation of Knapsack Functions, Operations Research 14, (1966), 1045–1074.CrossRefGoogle Scholar
  13. [13]
    D.S. Hirschberg, C.K. Wong, A polynomial time algorithm for the knapsack problem with two variables, JACM 23 (1976), 147–154.CrossRefGoogle Scholar
  14. [14]
    R. Kannan, A proof that integer programming is in NP, unpublished, (1976).Google Scholar
  15. [15]
    R. Kannan, A Polynomial Algorithm For the Two-Variable Integer Programming Problem, Tech. Report 348, Dept. Oper. Res., Cornell Univ., July 1977.Google Scholar
  16. [16]
    R.M. Karp, Reducibilities among combinatorial problems, in Complexity of Computer Computations, (eds. R.E. Miller, J.W. Thatcher), Plenum Press, (1972), 85–103.Google Scholar
  17. [17]
    R.M. Karp, On the computational complexity of combinatorial problems, Networks 5, (1975), 44–68.Google Scholar
  18. [18]
    G.S. Lueker, Two polynomial complete problems in nonnegative integer programming, TR-178, Dept. Computer Science, Princeton Univ., March 1975.Google Scholar
  19. [19]
    S. Sahni, Computationally related problems, SIAM J. Cpt. 3 (1974).Google Scholar
  20. [20]
    E. Specker, V. Strassen, Komplexitaet von Entscheidungsproblemen, Chapter IV by J.v.z. Gathen, M. Sieveking, Lecture Notes in Computer Science 43, Springer-Verlag, New York, 1976.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1978

Authors and Affiliations

  • Ravindran Kannan
    • 1
  • Clyde L. Monma
    • 2
  1. 1.University of BonnWest Germany
  2. 2.Cornell UniversityUSA

Personalised recommendations