Canonical Coin Systems for Change-Making Problems

The Change-Making Problem is to represent a given value with the fewest coins under a given coin system. As a variation of the knapsack problem, it is known to be NP-hard. Nevertheless, in most real money systems, the greedy algorithm yields optimal solutions. In this paper, we study what type of co...

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Bibliographic Details
Main Author Cai, Xuan
Format Journal Article
LanguageEnglish
Published 02.09.2008
Subjects
Computer Science - Data Structures and Algorithms
Computer Science - Discrete Mathematics
Online AccessGet full text
DOI10.48550/arxiv.0809.0400

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Summary:The Change-Making Problem is to represent a given value with the fewest coins under a given coin system. As a variation of the knapsack problem, it is known to be NP-hard. Nevertheless, in most real money systems, the greedy algorithm yields optimal solutions. In this paper, we study what type of coin systems that guarantee the optimality of the greedy algorithm. We provide new proofs for a sufficient and necessary condition for the so-called canonical coin systems with four or five types of coins, and a sufficient condition for non-canonical coin systems, respectively. Moreover, we present an$O(m^2)$algorithm that decides whether a tight coin system is canonical.
DOI:10.48550/arxiv.0809.0400