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 - https://doi.org/10.1007/s10107-021-01755-7
DO - https://doi.org/10.1007/s10107-021-01755-7
M3 - مقالة
SN - 0025-5610
VL - 198
SP - 27
EP - 83
JO - Mathematical Programming
JF - Mathematical Programming
IS - 1
ER -