A PTAS for the Time-Invariant Incremental Knapsack problem

TL;DR

This paper introduces a Polynomial-Time Approximation Scheme (PTAS) for the Time-Invariant Incremental Knapsack problem, a complex multi-period optimization challenge, by employing a novel rounding technique on a disjunctive formulation.

Contribution

It is the first to provide a PTAS for IIK, settling its complexity and extending the approach to related variants.

Findings

Developed a PTAS for IIK using disjunctive rounding

Proved the complexity of IIK is strongly NP-hard

Extended the technique to various problem variants

Abstract

The Time-Invariant Incremental Knapsack problem (IIK) is a generalization of Maximum Knapsack to a discrete multi-period setting. At each time, capacity increases and items can be added, but not removed from the knapsack. The goal is to maximize the sum of profits over all times. IIK models various applications including specific financial markets and governmental decision processes. IIK is strongly NP-hard and there has been work on giving approximation algorithms for some special cases. In this paper, we settle the complexity of IIK by designing a PTAS based on rounding a disjuncive formulation, and provide several extensions of the technique.