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# More on Change-Making and Related Problems

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LIPIcs.ESA.2020.29.pdf
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## Acknowledgements

We thank Adam Polak and Chao Xu for discussion and, in particular, for bringing the minimum word break problem to our attention.

## Cite As

Timothy M. Chan and Qizheng He. More on Change-Making and Related Problems. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.29

## 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(u²+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-u²-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).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• Coin changing
• knapsack
• dynamic programming
• Frobenius problem
• fine-grained complexity

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## References

1. Noga Alon, Zvi Galil, Oded Margalit, and Moni Naor. Witnesses for Boolean matrix multiplication and for shortest paths. In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 417-426, 1992. URL: https://doi.org/10.1109/SFCS.1992.267748.
2. Kyriakos Axiotis and Christos Tzamos. Capacitated dynamic programming: Faster knapsack and graph algorithms. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP), pages 19:1-19:13, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.19.
3. Arturs Backurs and Piotr Indyk. Which regular expression patterns are hard to match? In Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 457-466, 2016.
4. MohammadHossein Bateni, MohammadTaghi Hajiaghayi, Saeed Seddighin, and Cliff Stein. Fast algorithms for knapsack via convolution and prediction. In Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC), pages 1269-1282, 2018. URL: https://doi.org/10.1145/3188745.3188876.
5. Karl Bringmann. A near-linear pseudopolynomial time algorithm for subset sum. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1073-1084, 2017.
6. Karl Bringmann, Allan Grønlund, and Kasper Green Larsen. A dichotomy for regular expression membership testing. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 307-318, 2017. URL: http://arxiv.org/abs/1611.00918.
7. Timothy M. Chan. More algorithms for all-pairs shortest paths in weighted graphs. SIAM Journal on Computing, 39(5):2075-2089, 2010.
8. Timothy M. Chan and Qizheng He. On the change-making problem. In Proceedings of the 4th ACM-SIAM Symposium on Simplicity in Algorithms (SOSA), pages 38-42, 2020.
9. Marek Cygan, Marcin Mucha, Karol Wegrzycki, and Michal Wlodarczyk. On problems equivalent to (min, +)-convolution. ACM Transactions on Algorithms, 15(1):14:1-14:25, 2019. URL: https://doi.org/10.1145/3293465.
10. Jacques Dixmier. Proof of a conjecture by Erdős and Graham concerning the problem of Frobenius. Journal of Number Theory, 34(2):198-209, 1990.
11. Ran Duan and Seth Pettie. Fast algorithms for (max,min)-matrix multiplication and bottleneck shortest paths. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 384-391, 2009.
12. Friedrich Eisenbrand and Robert Weismantel. Proximity results and faster algorithms for integer programming using the Steinitz lemma. ACM Transactions on Algorithms, 16(1):5:1-5:14, 2020. URL: https://doi.org/10.1145/3340322.
13. Paul Erdős and Ronald L Graham. On a linear diophantine problem of Frobenius. Acta Arithmetica, 21(1):399-408, 1972.
14. Klaus Jansen and Lars Rohwedder. On integer programming and convolution. In Proceedings of the 10th Innovations in Theoretical Computer Science Conference (ITCS), pages 43:1-43:17, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.43.
15. Ce Jin and Hongxun Wu. A simple near-linear pseudopolynomial time randomized algorithm for subset sum. In Proceedings of the 2nd Symposium on Simplicity in Algorithms (SOSA), volume 69, pages 17:1-17:6, 2019.
16. Konstantinos Koiliaris and Chao Xu. A faster pseudopolynomial time algorithm for subset sum. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1062-1072, 2017.
17. Konstantinos Koiliaris and Chao Xu. Faster pseudopolynomial time algorithms for subset sum. ACM Transactions on Algorithms, 15(3):1-20, 2019.
18. Marvin Künnemann, Ramamohan Paturi, and Stefan Schneider. On the fine-grained complexity of one-dimensional dynamic programming. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP), pages 21:1-21:15, 2017.
19. Andrea Lincoln, Adam Polak, and Virginia Vassilevska Williams. Monochromatic triangles, intermediate matrix products, and convolutions. In Proceedings of the 11th Innovations in Theoretical Computer Science Conference (ITCS), pages 53:1-53:18, 2020.
20. George S. Lueker. Two NP-complete problems in nonnegative integer programming. Technical report, Princeton University. Department of Electrical Engineering, 1975.
21. Jiří Matoušek. Computing dominances in Eⁿ. Information Processing Letters, 38(5):277-278, 1991. URL: https://doi.org/10.1016/0020-0190(91)90071-O.
22. David Pisinger. Linear time algorithms for knapsack problems with bounded weights. Journal of Algorithms, 33(1):1-14, 1999.
23. Raimund Seidel. On the all-pairs-shortest-path problem. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing (STOC), pages 745-749, 1992. URL: https://doi.org/10.1145/129712.129784.
24. Arie Tamir. New pseudopolynomial complexity bounds for the bounded and other integer knapsack related problems. Operations Research Letters, 37(5):303-306, 2009. URL: https://doi.org/10.1016/j.orl.2009.05.003.
25. Virginia Vassilevska, R. Ryan Williams, and Raphael Yuster. All pairs bottleneck paths and max-min matrix products in truly subcubic time. Theory of Computing, 5(1):173-189, 2009.
26. R. Ryan Williams. Faster all-pairs shortest paths via circuit complexity. SIAM Journal on Computing, 47(5):1965-1985, 2018. URL: https://doi.org/10.1137/15M1024524.
27. J. W. Wright. The change-making problem. Journal of the ACM, 22(1):125-128, 1975.
28. Chao Xu. Word break with cost. https://chaoxuprime.com/posts/2019-09-19-word-break-with-cost.html, 2019.
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