More on Change-Making and Related Problems

TL;DR

This paper presents new algorithms and improvements for the change-making problem and related knapsack problems, achieving faster runtimes and extending applicability to all-targets and all-capacities versions.

Contribution

It introduces novel algorithms with improved time complexities for all-targets change-making, single-target change-making, and unbounded knapsack problems, extending previous work.

Findings

New $ ilde{O}(t^{4/3})$-time algorithm for all-targets change-making.

Simple $ ilde{O}(u^2+t)$-time algorithm for all-targets change-making.

Modified $ ilde{O}(u)$-time algorithm for single-target change-making.

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 $j=0,\ldots,t$. 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 $O(t\,\textrm{polylog}\,t)$-time algorithms for the original (single-target) version of the change-making problem, but not the all-targets version. We obtain a number of new results on change-making and related problems, including: 1. A new algorithm for the all-targets change-making problem with running time $\tilde{O}(t^{4/3})$, improving a previous $\tilde{O}(t^{3/2})$-time algorithm. 2. A very simple $\tilde{O}(u^2+t)$-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 Erd\H{o}s 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 $u$). 3. For the original (single-target) coin changing problem, we describe a simple modification of one of Chan and He's algorithms that runs in $\tilde{O}(u)$ time (instead of $\tilde{O}(t)$). 4. For the original (single-capacity) unbounded knapsack problem, we describe a simple algorithm that runs in $\tilde{O}(nu)$ time, improving previous near-$u^2$-time algorithms. 5. 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).