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

TL;DR

This paper characterizes canonical currency systems with six coin types, providing conditions under which the greedy algorithm guarantees optimal change-making solutions, extending previous work on systems with fewer coin types.

Contribution

It offers a new characterization of six-coin canonical systems and proposes a partial generalization of existing characterizations for fewer coin types.

Findings

Characterization of six-coin canonical systems

Necessary and sufficient conditions for greedy optimality

Extension of previous characterizations to more coin types

Abstract

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