## Abstract

We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of n items, each associated with a non-negative weight, and T time periods with non-decreasing capacities W_{1}≤ ⋯ ≤ W_{T}. When item i is inserted at time t, we gain a profit of p_{it}; however, this item remains in the knapsack for all subsequent periods. The goal is to decide if and when to insert each item, subject to the time-dependent capacity constraints, with the objective of maximizing our total profit. Interestingly, this setting subsumes as special cases a number of recently-studied incremental knapsack problems, all known to be strongly NP-hard. Our first contribution comes in the form of a polynomial-time (12-ϵ)-approximation for the generalized incremental knapsack problem. This result is based on a reformulation as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys–Tardos algorithm for the generalized assignment problem. Combined with further enumeration-based self-reinforcing ideas and new structural properties of nearly-optimal solutions, we turn our algorithm into a quasi-polynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.

Original language | English |
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Journal | Mathematical Programming |

DOIs | |

State | Accepted/In press - 2022 |

Externally published | Yes |

### Bibliographical note

Funding Information:Faenza and Zhang acknowledge funding from the ONR award N00014-20-1-2091. Danny Segev’s research on this project is supported by ISF Grants 148/16 and 1407/20.

Publisher Copyright:

© 2021, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

## Keywords

- Approximation algorithms
- Incremental optimization
- Sequencing

## ASJC Scopus subject areas

- Software
- Mathematics (all)