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

TL;DR

This paper revisits the cascading-tree branch-and-bound algorithm for the binary 0-1 Knapsack Problem, demonstrating its effectiveness through computational experiments and highlighting its historical significance in operations research.

Contribution

It reintroduces and evaluates a classical algorithm for the binary KP, showcasing its performance compared to modern methods and honoring Heiner Mülller-Merbach's legacy.

Findings

The cascading-tree algorithm is effective for binary KP.

Computational results favor the classical approach over some modern methods.

The algorithm's potential was previously overlooked due to limited literature.

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.