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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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 ( 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.
AbstractList	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 [Formula omitted]. When item i is inserted at time t, we gain a profit of [Formula omitted]; 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 [Formula omitted]-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. 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.
Audience	Academic
Author	Faenza, Yuri Segev, Danny Zhang, Lingyi
Author_xml	– sequence: 1 givenname: Yuri orcidid: 0000-0002-3148-2159 surname: Faenza fullname: Faenza, Yuri organization: Department of Industrial Engineering and Operations Research, Columbia University – sequence: 2 givenname: Danny orcidid: 0000-0003-4684-2185 surname: Segev fullname: Segev, Danny email: segevdanny@tauex.tau.ac.il organization: Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University – sequence: 3 givenname: Lingyi orcidid: 0000-0001-7562-4449 surname: Zhang fullname: Zhang, Lingyi organization: Department of Industrial Engineering and Operations Research, Columbia University
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References_xml	– reference: Chakaravarthy, V.T., Choudhury, A.R., Gupta, S., Roy, S., Sabharwal, Y.: Improved algorithms for resource allocation under varying capacity. In: Proceedings for the 22nd Annual European Symposium on Algorithms, pp. 222–234 (2014) – reference: HochbaumDSMaassWApproximation schemes for covering and packing problems in image processing and VLSIJ. ACM198532113013683233510.1145/2455.2141060633.68027 – reference: FleischerLGoemansMXMirrokniVSSviridenkoMTight approximation algorithms for maximum separable assignment problemsMath. Oper. Res.2011363416431283239910.1287/moor.1110.04991238.68187 – reference: ChekuriCKhannaSA polynomial time approximation scheme for the multiple knapsack problemSIAM J. Comput.2005353713728220145510.1137/S00975397003828201095.68035 – reference: Bansal, N., Chakrabarti, A., Epstein, A., Schieber, B.: A quasi-PTAS for unsplittable flow on line graphs. In: Proceedings of the 38th ACM Symposium on Theory of Computing, pp. 721–729 (2006) – reference: SkutellaMCookWLovászLVygenJAn introduction to network flows over timeResearch Trends in Combinatorial Optimization2009BerlinSpringer45148210.1007/978-3-540-76796-1_21 – reference: GrafLHarksTSeringLDynamic flows with adaptive route choiceMath. Program.2020183309335413782210.1007/s10107-020-01504-21448.90024 – reference: FeigeUVondrákJThe submodular welfare problem with demand queriesTheory Comput.20106247290277007810.4086/toc.2010.v006a0111213.68703 – reference: Caprara, A.: Packing 2-dimensional bins in harmony. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, pp. 490–499 (2002) – reference: Grandoni, F., Mömke, T., Wiese, A., Zhou, H.: A (5/3+ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5/3+\epsilon $$\end{document})-approximation for unsplittable flow on a path: placing small tasks into boxes. In: Proceedings of the 50th Annual ACM Symposium on Theory of Computing, pp. 607–619 (2018) – reference: Hartline, J.R.K.: Incremental optimization. Ph.D. Thesis, Department of Computer Science, Cornell University (2008) – reference: Ye, C.: On the trade-offs between modeling power and algorithmic complexity. 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Snippet	We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack . In this setting, we are... We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are...
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Title	Approximation algorithms for the generalized incremental knapsack problem
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