Skip to main content

The Knapsack Problem

  • Chapter
Combinatorial Optimization

Part of the book series: Algorithms and Combinatorics ((AC,volume 21))

  • 4511 Accesses

Abstract

The Minimum Weight Perfect Matching Problem and the Weighted Matroid Intersection Problem discussed in earlier chapters are among the “hardest” problems for which a polynomial-time algorithm is known. In this chapter we deal with the following problem which turns out to be, in a sense, the “easiest” NP-hard problem:

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

General Literature

  • Garey, M.R., and Johnson, D.S. [1979]: Computers and Intractability; A Guide to the Theory of NP-Completeness. Freeman, San Francisco 1979, Chapter 4

    Google Scholar 

  • Martello, S., and Toth, P. [1990]: Knapsack Problems; Algorithms and Computer Implementations. Wiley, Chichester 1990

    MATH  Google Scholar 

  • Papadimitriou, C.H., and Steiglitz, K. [1982]: Combinatorial Optimization; Algorithms and Complexity. Prentice-Hall, Englewood Cliffs 1982, Sections 16.2, 17.3, and 17.4

    MATH  Google Scholar 

Cited References

  • Bellman, R. [1956]: Notes on the theory of dynamic programming IV — maximization over discrete sets. Naval Research Logistics Quarterly 3 (1956), 67–70

    Article  MathSciNet  Google Scholar 

  • Bellman, R. [1957]: Comment on Dantzig’s paper on discrete variable extremum problems. Operations Research 5 (1957), 723–724

    Article  Google Scholar 

  • Blum, M., Floyd, R.W., Pratt, V., Rivest, R.L., and Tarjan, R.E. [1973]: Time bounds for selection. Journal of Computer and System Sciences 7 (1973), 448–461

    Article  MATH  MathSciNet  Google Scholar 

  • Dantzig, G.B. [1957]: Discrete variable extremum problems. Operations Research 5 (1957), 266–277

    MathSciNet  Google Scholar 

  • Garey, M.R., and Johnson, D.S. [1978]: Strong NP-completeness results: motivation, examples, and implications. Journal of the ACM 25 (1978), 499–508

    Article  MATH  MathSciNet  Google Scholar 

  • Gens, G.V., and Levner, E.V. [1979]: Computational complexity of approximation algorithms for combinatorial problems. In: Mathematical Foundations of Computer Science; LNCS 74 (J. Becvar, ed.), Springer, Berlin 1979, pp. 292–300

    Google Scholar 

  • Ibarra, O.H., and Kim, C.E. [1975]: Fast approximation algorithms for the knapsack and sum of subset problem. Journal of the ACM 22 (1975), 463–468

    Article  MATH  MathSciNet  Google Scholar 

  • Kellerer, H., and Pferschy, U. [2004]: Improved dynamic programming in connection with an FPTAS for the knapsack problem. Journal on Combinatorial Optimization 8 (2004), 5–11

    Article  MATH  MathSciNet  Google Scholar 

  • Korte, B., and Schrader, R. [1981]: On the existence of fast approximation schemes. In: Nonlinear Programming; Vol. 4 (O. Mangaserian, R.R. Meyer, S.M. Robinson, eds.), Academic Press, New York 1981, pp. 415–437

    Google Scholar 

  • Lawler, E.L. [1979]: Fast approximation algorithms for knapsack problems. Mathematics of Operations Research 4 (1979), 339–356

    MATH  MathSciNet  Google Scholar 

  • Pisinger, D. [1999]: Linear time algorithms for knapsack problems with bounded weights. Journal of Algorithms 33 (1999), 1–14

    Article  MATH  MathSciNet  Google Scholar 

  • Sahni, S. [1976]: Algorithms for scheduling independent tasks. Journal of the ACM 23 (1976), 114–127

    Google Scholar 

  • Vygen, J. [1997]: The two-dimensional weighted median problem. Zeitschrift für Angewandte Mathematik und Mechanik 77 (1997), Supplement, S433–S436

    MATH  Google Scholar 

  • Woeginger, G.J. [2000]: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? INFORMS Journal on Computing 12 (2000), 57–74

    Article  MATH  MathSciNet  Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

(2008). The Knapsack Problem. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71844-4_17

Download citation

Publish with us

Policies and ethics