More on change-making and related problems.

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. We obtain a number of new results on these problems, including (i) an algorithm for the all-targets problem with running time O ˜ (t 4 / 3) (improving a previous O ˜ (t 3 / 2) -time algorithm), (ii) a very simple O ˜ (u 2 + t) -time algorithm for the all-targets problem, where u denotes the maximum coin value, and (iii) an algorithm for the original (single-target) problem with running time O ˜ (u) (improving a previous O ˜ (t) -time algorithm by Chan and He (SOSA'20)). We also describe applications to the (single-capacity) unbounded knapsack problem and the minimum word break problem. [ABSTRACT FROM AUTHOR]