Optimal bounds for the change-making problem
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  show that if the greedy algorithm is not always optimal, then there exists a counterexample x in the range c3 ≤ x < 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.
Unable to display preview. Download preview PDF.
- 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
- 2.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.G. S. Lueker. Two NP-complete problems in nonnegative integer programming. Technical Report 178, Computer Science Laboratory, Princeton University, 1975.Google Scholar
- 4.S. Martello and P. Toth. Knapsack Problems. John Wiley and Sons, 1990.Google Scholar