Mathematical Programming

, Volume 45, Issue 1–3, pp 295–309 | Cite as

On the supermodular knapsack problem

  • G. Gallo
  • B. Simeone


In this paper we introduce binary knapsack problems where the objective function is nonlinear, and investigate their Lagrangean and continuous relaxations. Some of our results generalize previously known theorems concerning linear and quadratic knapsack problems. We investigate in particular the case in which the objective function is supermodular. Under this hypothesis, although the problem remains NP-hard, we show that its Lagrangean dual and its continuous relaxation can be solved in polynomial time. We also comment on the complexity of recognizing supermodular functions. The particular case in which the knapsack constraint is of the cardinality type is also addressed and some properties of its optimal value as a function of the right hand side are derived.


Objective Function Mathematical Method Polynomial Time Knapsack Problem Continuous Relaxation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. M.S. Bazaraa and C.M. Shetty,Nonlinear Programming, Theory and Algorithms (Wiley, New York, 1979).Google Scholar
  2. A. Billionnet and M. Minoux, “Maximizing a supermodular pseudoboolean function: a polynomial algorithm for supermodular cubic functions,”Discrete Applied Mathematics 12 (1985) 1–11.Google Scholar
  3. P. Chaillou, P. Hansen and Y. Mahieu, “Best network flow bounds for the quadratic knpsack problem,” presented at NETFLOW83 International Workshop, Pisa, Italy (1983).Google Scholar
  4. Y. Crama, “Recognition problems for special classes of pseudoboolean functions,” RUTCOR Research Report, Rutgers University (New Brunswick, NJ, 1986).Google Scholar
  5. M.L. Fisher, G.L. Nemhauser and L.A. Wolsey, “An analysis of approximations for maximizing submodular set functions—I,”Mathematical Programming 14 (1978) 265–294.Google Scholar
  6. G. Gallo, M.D. Grigoriadis and R.E. Tarjan, “A parametric maximum flow algorithm,” Department of Computer Science, Rutgers University (New Brunswick, NY, 1987).Google Scholar
  7. G. Gallo, P. Hammer and B. Simeone, “Quadratic knapsack problems,”Mathematical Programming 12 (1980) 132–149.Google Scholar
  8. A.V. Goldberg and R.E. Tarjan, “A new approach to the maximum flow problem,”Proceedings of the 18th Annual ACM Symposium on Theory of Computing (1986) pp. 136–146.Google Scholar
  9. M. Grötschel, L. Lovasz and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization,”Combinatorica 1 (1981) 169–197.Google Scholar
  10. M. Grötschel, L. Lovasz and A. Schrijver,Geometric Algorithms in Combinatorial Optimization (Springer, New York, 1988).Google Scholar
  11. P.L. Hammer and S. Rudeanu,Boolean methods in Operations Research and related areas (Springer, Heidelberg, 1968).Google Scholar
  12. P.L. Hammer and B. Simeone, “Quasimonotone boolean functions and bistellar graphs,”Annals of Discrete Mathematics 9 (1980) 107–119.Google Scholar
  13. R.M. Karp, “Reducibility among combinatorial problems,” in: R.E. Miller and J.W. Thatcher, eds.,Complexity of Computer Computations (Plenum Press, New York, 1972).Google Scholar
  14. B. Simeone, “Quadratic 0–1 programming, Boolean functions and graphs,” Doctoral Dissertation, Waterloo University (1979).Google Scholar
  15. D.A. Topkis, “Minimizing a submodular function on a lattice,”Operations Research 26 (1978) 305–321.Google Scholar
  16. C. Witzgall, “Mathematical methods of site selection for Electronic Message Systems (EMS),” NBS Internal report (1975).Google Scholar

Copyright information

© North-Holland 1989

Authors and Affiliations

  • G. Gallo
    • 1
  • B. Simeone
    • 2
  1. 1.Dipartimento di InformaticaUniversità di PisaPisaItaly
  2. 2.Dipartimento de StatisticaUniversità di Roma “La Sapienza”RomeItaly

Personalised recommendations