Approximation algorithms for the generalized incremental knapsack problem

Yuri Faenza, Danny Segev, Lingyi Zhang

Research output: Contribution to journalArticlepeer-review


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 W1≤ ⋯ ≤ WT. When item i is inserted at time t, we gain a profit of pit; 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 languageEnglish
Pages (from-to)27-83
Number of pages57
JournalMathematical Programming
Issue number1
StatePublished - Mar 2023
Externally publishedYes

Bibliographical note

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


  • Approximation algorithms
  • Incremental optimization
  • Sequencing

ASJC Scopus subject areas

  • Software
  • General Mathematics


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