General Knapsack Problems in a Dynamic Setting

TL;DR

This paper introduces a PTAS for the Generalized Multistage d-Knapsack problem, addressing the dynamic nature of real-world optimization by balancing solution quality and stability over time.

Contribution

It extends multistage knapsack models to a generalized setting and provides a polynomial-time approximation scheme for it.

Findings

Developed a PTAS for the generalized multistage d-Knapsack.

Addresses dynamic changes and solution stability in optimization.

Extends multistage models to more general knapsack problems.

Abstract

The world is dynamic and changes over time, thus any optimization problem used to model real life problems must address this dynamic nature, taking into account the cost of changes to a solution over time. The multistage model was introduced with this goal in mind. In this model we are given a series of instances of an optimization problem, corresponding to different times, and a solution is provided for each instance. The strive for obtaining near-optimal solutions for each instance on one hand, while maintaining similar solutions for consecutive time units on the other hand, is quantified and integrated into the objective function. In this paper we consider the Generalized Multistage $d$-Knapsack problem, a generalization of the multistage variants of the Multiple Knapsack problem, as well as the $d$-Dimensional Knapsack problem. We present a PTAS for Generalized Multistage $d$-Knapsack.