## Abstract

Given a set of n integer-valued coin types and a target value t, the well-known change-making problem asks for the minimum number of coins that sum to t, assuming an unlimited number of coins in each type. In the more general all-targets version of the problem, we want the minimum number of coins summing to j, for every j = 0, . . ., t. For example, the textbook dynamic programming algorithms can solve the all-targets problem in O(nt) time. Recently, Chan and He (SOSA’20) described a number of O(t polylog t)-time algorithms for the original (single-target) version of the change-making problem, but not the all-targets version. In this paper, we obtain a number of new results on change-making and related problems: We present a new algorithm for the all-targets change-making problem with running time Õ(t^{4}/^{3}), improving a previous Õ(t^{3}/^{2})-time algorithm. We present a very simple Õ(u^{2} + t)-time algorithm for the all-targets change-making problem, where u denotes the maximum coin value. The analysis of the algorithm uses a theorem of Erdős and Graham (1972) on the Frobenius problem. This algorithm can be extended to solve the all-capacities version of the unbounded knapsack problem (for integer item weights bounded by u). For the original (single-target) coin changing problem, we describe a simple modification of one of Chan and He’s algorithms that runs in Õ(u) time (instead of Õ(t)). For the original (single-capacity) unbounded knapsack problem, we describe a simple algorithm that runs in Õ(nu) time, improving previous near-u^{2}-time algorithms. We also observe how one of our ideas implies a new result on the minimum word break problem, an optimization version of a string problem studied by Bringmann et al. (FOCS’17), generalizing change-making (which corresponds to the unary special case).

Original language | English (US) |
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Title of host publication | 28th Annual European Symposium on Algorithms, ESA 2020 |

Editors | Fabrizio Grandoni, Grzegorz Herman, Peter Sanders |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771627 |

DOIs | |

State | Published - Aug 1 2020 |

Event | 28th Annual European Symposium on Algorithms, ESA 2020 - Virtual, Pisa, Italy Duration: Sep 7 2020 → Sep 9 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 173 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 28th Annual European Symposium on Algorithms, ESA 2020 |
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Country/Territory | Italy |

City | Virtual, Pisa |

Period | 9/7/20 → 9/9/20 |

## Keywords

- Coin changing
- Dynamic programming
- Fine-grained complexity
- Frobenius problem
- Knapsack

## ASJC Scopus subject areas

- Software