Combinatorics of the change-making problem

TL;DR

This paper explores the structure of coin systems where the greedy algorithm always yields the minimum number of coins, providing necessary conditions and relations between such systems and their sub-systems.

Contribution

It introduces necessary conditions for coin systems that guarantee the greedy algorithm's optimality and examines their relationships with sub-currencies.

Findings

Identifies necessary conditions for coin systems with optimal greedy solutions

Establishes relations between such currencies and their sub-currencies

Provides insights into the structure of change-making systems

Abstract

We investigate the structure of the currencies (systems of coins) for which the greedy change-making algorithm always finds an optimal solution (that is, a one with minimum number of coins). We present a series of necessary conditions that must be satisfied by the values of coins in such systems. We also uncover some relations between such currencies and their sub-currencies.