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 = 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 c₃ ≤ x < c_m(c _m c _m−1+ c _m − 3cm1/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 c₃+ 1 <x < c _m + c _{m− 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.