Optimal bounds for the change-making problem

Abstract

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= c 1< c 2<⋯< c m . Chang and Gill (1970) show that if the greedy algorithm is not always optimal, then there exists a counterexample x in the range c 3≤x< c m(c mc m−1+c m−3c m−1) c m−c m−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 c 3+1 <x<c m+c m−1 .