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 prob lem of determining whether the greedy algorithm produces an opti mal representation of all amounts for a given set of coin denominations c c cm Chang and Gill show that if the greedy algorithm is not always optimal then there exists a counterexample x in the range c x cm cmcm cm cm cm cm To test for the existence of such a counterexample Chang and Gill pro pose 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 c x cm cm and these bounds are tight Moreover we give a simple test for the ex istence of a counterexample that does not require the calculation of op timal representations In addition we give a complete characterization of three coin systems and an e cient algorithm for all systems with a xed number of coins Finally we show that a related problem is coNP complete