Approximation algorithms for the generalized incremental knapsack problem

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 ite...

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Veröffentlicht in:Mathematical programming Jg. 198; H. 1; S. 27 - 83
Hauptverfasser: Faenza, Yuri, Segev, Danny, Zhang, Lingyi
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2023
Springer
Schlagworte:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
90C10
Approximation algorithms
68W25
Sequencing
Incremental optimization
90B35
ISSN:0025-5610, 1436-4646
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Zusammenfassung: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 ( 1 2 - 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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-021-01755-7