Optimal bounds for the change-making problem

  • Dexter Kozen
  • Shmuel Zaks
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 700)


The change-making problem is the problem of representing a given value with the fewest coins possible. We investigate the problem of determining whether the greedy algorithm produces an optimal representation of all amounts for a given set of coin denominations 1 = c1 < c2 < ... < c m . Chang and Gill [1] show that if the greedy algorithm is not always optimal, then there exists a counterexample x in the range c3x < cm(c m cm−1+ c m − 3cm1/cm−cm− 1.

To test for the existence of such a counterexample, Chang and Gill propose computing and comparing the greedy and optimal representations of all x in this range. In this paper we show that if a counterexample exists, then the smallest one lies in the range c3+ 1 <x < c m + cm− 1, and these bounds are tight. Moreover, we give a simple test for the existence of a counterexample that does not require the calculation of optimal representations. In addition, we give a complete characterization of three-coin systems and an efficient algorithm for all systems with a fixed number of coins. Finally, we show that a related problem is cqNP- complete.


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  1. 1.
    S. K. Chang and A. Gill. Algorithmic solution of the change-making problem. J. Assoc. Comput. Mach., 17(1):113–122, January 1970.Google Scholar
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    M. R. Garey and D. S. Johnson. Computers and Intractability: a Guide to the Theory of NP-Completeness. W. H. Freeman, 1979.Google Scholar
  3. 3.
    G. S. Lueker. Two NP-complete problems in nonnegative integer programming. Technical Report 178, Computer Science Laboratory, Princeton University, 1975.Google Scholar
  4. 4.
    S. Martello and P. Toth. Knapsack Problems. John Wiley and Sons, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Dexter Kozen
    • 1
  • Shmuel Zaks
    • 2
  1. 1.Computer Science DepartmentCornell UniversityIthacaUSA
  2. 2.Computer Science DepartmentTechnionHaifaIsrael

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