Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2111.12392

Characterization of canonical systems with six types of coins for the change-making problem

Yuma Suzuki·Ryuhei Miyashiro
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper analyzes a necessary and sufficient condition for the change-making problem to be solvable with a greedy algorithm. The change-making problem is to minimize the number of coins used to pay a given value in a specified currency system. This problem is NP-hard, and therefore the greedy algorithm does not always yield an optimal solution. Yet for almost all real currency systems, the greedy algorithm outputs an optimal solution. A currency system for which the greedy algorithm returns an optimal solution for any value of payment is called a canonical system. Canonical systems with at most five types of coins have been characterized in previous studies. In this paper, we give characterization of canonical systems with six types of coins, and we propose a partial generalization of characterization of canonical systems.

Explore related subjects

cs.DM

Keep this discovery

BibTeXRIS

Yuma Suzuki, Ryuhei Miyashiro. 2021-11-24. Characterization of canonical systems with six types of coins for the change-making problem. https://arxiv.org/abs/2111.12392

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An FPTAS for Two-Machine Open-Shop Scheduling with a Single Unavailability Interval

We consider the two-machine open-shop scheduling problem in which one machine is unavailable during a fixed interval. We study the resumable setting: an operation interrupted by the unavailability interval may resume, without penalty, when the machine becomes available. The objective is to minimize the makespan. Although the problem is NP-hard and several approximation algorithms are known, whether it admits a fully polynomial-time approximation scheme (FPTAS) has remained open for two decades. We resolve this question affirmatively by giving the first FPTAS, thereby strengthening the previously known polynomial-time approximation scheme (PTAS). As an intermediate result, we develop a new pseudo-polynomial dynamic program with seven state dimensions, improving on the ten-dimensional formulation in the literature.

cs.DM↗
arXiv2026

Generalized Graph Search Trees

Graph search algorithms and their corresponding graph search trees are commonly used in algorithmic graph theory. In recent years, the recognition problem of these graph search trees has received significant attention. So far, the research has focused on two types of search trees: first-in trees that behave like BFS-trees and last-in trees that behave like DFS-trees. The search tree paradigms differ from each other by the parent a vertex is connected to. In first-in trees, it is the first visited neighbor, while in last-in trees it is the last neighbor visited before that vertex. Here, we will generalize these concepts of graph search trees by allowing every preceding neighbor of a vertex to be the parent. We study the complexity of the recognition problem of these generalized graph search trees. We present NP-completeness proofs for most searches. We also show that the problem is trivial for Generic Search and polynomial-time solvable for several searches on bipartite graphs and chordal graphs. We also study the question how fixing the start vertex influences the complexity of the problem.

cs.DM↗
arXiv2026

The exact asymptotic constant in the metric dimension of Jaccard space

Let $X$ be a finite set with $|X|=n$ and let $\mathrm{Jac}(a,b)=|a\,\triangle\, b|/|a\cup b|$ be the Jaccard distance on the power set $2^X$. Lladser and Paradise recently proved that the metric dimension of $(2^X,\mathrm{Jac})$ is $\Theta(n/\ln n)$, with the constant left open; their bounds are $(\ln 2)\,n/\ln n\lesssim \beta(2^X,\mathrm{Jac})\lesssim 2\ln(2e)\,n/\ln n$. We determine the constant: \[ \beta(2^X,\mathrm{Jac})=\frac{2n}{\log_2 n}\,(1+o(1))=(2\ln 2)\,\frac{n}{\ln n}\,(1+o(1)). \] The proof identifies the problem, on each ``slice'' of subsets of fixed cardinality, with the Erd\H{o}s--R\'enyi coin-weighing problem for a spring scale (the problem of \emph{detecting matrices}). The lower bound is the Erd\H{o}s--R\'enyi entropy argument applied to the middle slice; the upper bound follows from the explicit detecting families of Lindstr\"om and of Cantor and Mills, augmented by a single extra landmark that reveals cardinality.

cs.DM↗