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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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
02.09.2008
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.0809.0400 |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Cai, Xuan |
| Author_xml | – sequence: 1 givenname: Xuan surname: Cai fullname: Cai, Xuan |
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| Snippet | 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... |
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| SubjectTerms | Computer Science - Data Structures and Algorithms Computer Science - Discrete Mathematics |
| Title | Canonical Coin Systems for Change-Making Problems |
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