Technical Note - An Approximate Dynamic Programming Approach to The Incremental Knapsack Problem

Aouad, A and Segev, D (2023) Technical Note - An Approximate Dynamic Programming Approach to The Incremental Knapsack Problem. Operations Research, 71 (4). pp. 1414-1433. ISSN 0030-364X OPEN ACCESS


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. While various approximation algorithms were developed under mitigating structural assumptions, obtaining non-trivial 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.

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Item Type: Article
Subject Areas: Management Science and Operations
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© 2022 INFORMS

Date Deposited: 23 May 2022 12:08
Date of first compliant deposit: 02 Mar 2022
Subjects: Dynamic programming
Last Modified: 04 Jul 2024 09:26

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