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

Ali Aouad () and Danny Segev ()
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Ali Aouad: Department of Management Science and Operations, London Business School, London NW1 4SA, United Kingdom
Danny Segev: Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Operations Research, 2023, vol. 71, issue 4, 1414-1433

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.

Keywords: Optimization; incremental knapsack; approximate dynamic programming; PTAS (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:inm:oropre:v:71:y:2023:i:4:p:1414-1433

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