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This paper analyzes a necessary and sufficient condition for thechange-making problem to be solvable with a greedy algorithm. The change-makingproblem is to minimize the number of coins used to pay a given value in aspecified currency system. This problem is NP-hard, and therefore the greedyalgorithm does not always yield an optimal solution. Yet for almost all realcurrency systems, the greedy algorithm outputs an optimal solution. A currencysystem for which the greedy algorithm returns an optimal solution for any valueof payment is called a canonical system. Canonical systems with at most fivetypes of coins have been characterized in previous studies. In this paper, wegive characterization of canonical systems with six types of coins, and wepropose a partial generalization of characterization of canonical systems.

Characterization of canonical systems with six types of coins for the change-making problem