Technical Note—An Approximate Dynamic Programming Approach to the Incremental Knapsack Problem

Ali Aouad*, Danny Segev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We study the incremental knapsack problem, where one wishes to sequentially pack items into a knapsack whose capacity expands over a finite planning horizon, with the objective of maximizing time-averaged profits. Although various approximation algorithms were developed under mitigating structural assumptions, obtaining nontrivial performance guarantees for this problem in its utmost generality has remained an open question thus far. In this paper, we devise a polynomial-time approximation scheme for general instances of the incremental knapsack problem, which is the strongest guarantee possible given existing hardness results. In contrast to earlier work, our algorithmic approach exploits an approximate dynamic programming formulation. Starting with a simple exponentially sized dynamic program, we prove that an appropriate composition of state pruning ideas yields a polynomially sized state space with negligible loss of optimality. The analysis of this formulation synthesizes various techniques, including new problem decompositions, parsimonious counting arguments, and efficient rounding methods, that may be of broader interest.

Original languageEnglish
Pages (from-to)1414-1433
Number of pages20
JournalOperations Research
Volume71
Issue number4
DOIs
StatePublished - 1 Jul 2023

Keywords

  • PTAS
  • approximate dynamic programming
  • incremental knapsack

Fingerprint

Dive into the research topics of 'Technical Note—An Approximate Dynamic Programming Approach to the Incremental Knapsack Problem'. Together they form a unique fingerprint.

Cite this