A PTAS for the Time-Invariant Incremental Knapsack Problem

Faenza, Yuri; Malinović, Igor

doi:10.1007/978-3-319-96151-4_14

Cited by 10 publications

(7 citation statements)

References 20 publications

(32 reference statements)

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“…For instance, well-informed households generally wish to optimize how their cumulative income is invested over time into low-volatility securities that procure regular payments; from that angle, time-varying discounts reflect specific households' investment profiles. Similarly, this approach can be utilized for modeling the selection of investments in bond markets and various government decision processes, as illustrated by Faenza and Malinovic (2018). Additionally, the incremental knapsack problem has certain structural similarities with sequential assortment optimization problems in revenue management (Davis et al 2015, Gallego et al 2020, Aouad and Segev 2020, Derakhshan et al 2018) as well as with the generalized assignment problem (Shmoys and Tardos 1993, Chekuri and Khanna 2005, Fleischer et al 2011, Feige and Vondrák 2010.…”

Section: Introductionmentioning

confidence: 99%

“…In this setting, Bienstock et al (2013) attained a PTAS in the regime where the number of time periods is sublogarithmic in the number of items, i.e., T = O( √ log n). Recently, Faenza and Malinovic (2018) developed a PTAS for the time-invariant incremental knapsack problem without additional assumptions. Their approach combines techniques such as problem sparsification, compact enumeration, and LP-rounding methods.…”

Section: Introductionmentioning

confidence: 99%

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Technical Note—An Approximate Dynamic Programming Approach to the Incremental Knapsack Problem

Aouad

Segev

2023

Operations Research

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Integer packing problems have traditionally been some of the most fundamental and well-studied computational questions in discrete optimization. The paper by Aouad and Segev studies the incremental knapsack problem, where one wishes to sequentially pack items into a knapsack whose capacity expands over a finite planning horizon, with the objective of maximizing time-averaged profits. Although various approximation algorithms were developed under mitigating structural assumptions, obtaining nontrivial performance guarantees for this problem in its utmost generality has remained an open question thus far. The authors devise the first polynomial-time approximation scheme for general instances of the incremental knapsack problem, which is the strongest guarantee possible given existing hardness results. Their approach synthesizes various techniques related to approximate dynamic programming, including problem decompositions, counting arguments, and efficient rounding methods, which may be of broader interest.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Technical Note—An Approximate Dynamic Programming Approach to the Incremental Knapsack Problem

Aouad

Segev

2023

Operations Research

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“…A PTAS for the IKP when the discount factor is 1 (time invariant, referred to as IIKP) and T = O √ log n has been found in Bienstock et al (2013), and it has been shown that IIKP is strongly NP-hard. Later, Faenza and Malinovic (2018) proposed the first PTAS for IIKP regardless of T , and Della Croce et al ( 2019) proposed an PTAS for IKP when T is a constant. Most recent developments of IKP include Aouad and Segev (2020); Faenza et al (2020).…”

Section: Introductionmentioning

confidence: 99%

Approximation Schemes for Multiperiod Binary Knapsack Problems

Gao

Birge

Gupta

2021

Preprint

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An instance of the multiperiod binary knapsack problem (MPBKP) is given by a horizon length T , a nondecreasing vector of knapsack sizes (c 1 , . . . , c T ) where c t denotes the cumulative size for periods 1, . . . , t, and a list of n items. Each item is a triple (r, q, d) where r denotes the reward or value of the item, q its size, and d denotes its time index (or, deadline). The goal is to choose, for each deadline t, which items to include to maximize the total reward, subject to the constraints that for all t = 1, . . . , T , the total size of selected items with deadlines at most t does not exceed the cumulative capacity of the knapsack up to time t. We also consider the multiperiod binary knapsack problem with soft capacity constraints (MPBKP-S) where the capacity constraints are allowed to be violated by paying a penalty that is linear in the violation. The goal of MPBKP-S is to maximize the total profit, which is the total reward of the selected items less the total penalty. Finally, we consider the multiperiod binary knapsack problem with soft stochastic capacity constraints (MPBKP-SS), where the non-decreasing vector of knapsack sizes (c 1 , . . . , c T ) follow some arbitrary joint distribution but we are given access to the profit as an oracle, and we must choose a subset of items to maximize the total expected profit, which is the total reward less the total expected penalty.For MPBKP, we exhibit a fully polynomial-time approximation scheme that achieves (1 + ǫ) approximation with runtime Õ min n + T 3.25 ǫ 2.25 , n + T 2 ǫ 3 , nT ǫ 2 , n 2 ǫ ; for MPBKP-S, the (1 + ǫ) approximation can be achieved in O n log n ǫ• min T ǫ , n . To the best of our knowledge, our algorithms are the first FPTAS for any multiperiod version of the Knapsack problem since its study began in 1980s. For MPBKP-SS, we prove that a natural greedy algorithm is a 2-approximation when items have the same size. Our algorithms also provide insights on how other multiperiod versions of the knapsack problem may be approximated.

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“…Surprisingly, unlike the basic knapsack problem, Bienstock et al (2013) showed that this extension is strongly NP-hard. On the positive side, Faenza and Malinovic (2018) proposed a polynomial-time approximation scheme (PTAS) based on rounding fractional solutions to an appropriate disjunctive relaxation. In the broader incremental knapsack problem, we have p it = φ i • T τ =t ∆ τ , where ∆ τ ≥ 0 is a time-dependent scaling factor; in this context, Aouad and Segev (2020) have very recently obtained a PTAS, leveraging approximate dynamic programming ideas.…”

Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for The Generalized Incremental Knapsack Problem

Faenza¹,

Segev²,

Zhang³

2020

Preprint

Self Cite

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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 − ǫ)-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 newly-revealed structural properties of nearly-optimal solutions, we turn our basic algorithm into a quasipolynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.

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A PTAS for the Time-Invariant Incremental Knapsack Problem

Cited by 10 publications

References 20 publications

Technical Note—An Approximate Dynamic Programming Approach to the Incremental Knapsack Problem

Technical Note—An Approximate Dynamic Programming Approach to the Incremental Knapsack Problem

Approximation Schemes for Multiperiod Binary Knapsack Problems

Approximation Algorithms for The Generalized Incremental Knapsack Problem

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