TY - JOUR

T1 - Approximation algorithms for the generalized incremental knapsack problem

AU - Faenza, Yuri

AU - Segev, Danny

AU - Zhang, Lingyi

N1 - Publisher Copyright:
© 2021, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

PY - 2023/3

Y1 - 2023/3

N2 - 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.

AB - 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.

KW - Approximation algorithms

KW - Incremental optimization

KW - Sequencing

UR - http://www.scopus.com/inward/record.url?scp=85123857730&partnerID=8YFLogxK

U2 - 10.1007/s10107-021-01755-7

DO - 10.1007/s10107-021-01755-7

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AN - SCOPUS:85123857730

SN - 0025-5610

VL - 198

SP - 27

EP - 83

JO - Mathematical Programming

JF - Mathematical Programming

IS - 1

ER -