An Approximate Dynamic Programming Approach to The Incremental Knapsack Problem

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. 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.

Approximation Schemes for Multiperiod Binary Knapsack Problems

“…Later, Faenza and Malinovic (2018) proposed the first PTAS for IIKP regardless of T , and Della Croce et al ( 2019) proposed an PTAS for IKP when T is a constant. Most recent developments of IKP include Aouad and Segev (2020); Faenza et al (2020). Other similar problems and/or further extensions include the multiple-choice multiperiod knapsack problem (Lin and Chen, 2010;Lin and Wu, 2004;Randeniya, 1994), the multiperiod multi-dimensional knapsack problem (Lau and Lim, 2004), the multiperiod precedence-constrained knapsack problem (Moreno et al, 2010;Samavati et al, 2017), to name a few.…”