Cascading-Tree Algorithm for the 0-1 Knapsack Problem (In Memory of Heiner M{\"u}ller-Merbach, a Former President of IFORS)

Mahdi Moeini; Daniel Schermer; Oliver Wendt

Home/Archive/arXiv

arXiv · 2405.13450

Cascading-Tree Algorithm for the 0-1 Knapsack Problem (In Memory of Heiner M{\"u}ller-Merbach, a Former President of IFORS)

Mahdi Moeini·Daniel Schermer·Oliver Wendt

View original source ↗Read source document ↗Export metadata ↓

Abstract

In operations research, the Knapsack Problem (KP) is one of the classical optimization problems that has been widely studied. The KP has several variants and, in this paper, we address the binary KP, where for a given knapsack (with limited capacity) as well as a number of items, each of them has its own weight (volume or cost) and value, the objective consists in finding a selection of items such that the total value of the selected items is maximized and the capacity limit of the knapsack is respected. In this paper, in memorial of Prof. Dr. Heiner M{\"u}ller-Merbach, a former president of IFORS, we address the binary KP and revisit a classical algorithm, named cascading-tree branch-and-bound algorithm, that was originally introduced by him in 1978. However, the algorithm is surprisingly absent from the scientific literature because the paper was published in a German journal. We carried out computational experiments in order to compare the algorithm versus some classic methods. The numerical results show the effectiveness of the interesting idea used in the cascading-tree algorithm.

Explore related subjects

cs.DS math.OC

Keep this discovery

BibTeX RIS

Mahdi Moeini, Daniel Schermer, Oliver Wendt. 2024-05-22. Cascading-Tree Algorithm for the 0-1 Knapsack Problem (In Memory of Heiner M{\"u}ller-Merbach, a Former President of IFORS). https://arxiv.org/abs/2405.13450

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

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗