An Asymptotic Fully Polynomial Time Approximation Scheme for Bin Covering

Jansen, Klaus; Solis-Oba, Roberto

doi:10.1007/3-540-36136-7_16

Cited by 12 publications

(36 citation statements)

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“…Our problem of packing resizable items is closely related to the bin covering (BC) problem, which was widely studied (see, e.g., [1,5,6,13]). The input for bin covering is a set of items {a 1 , .…”

Section: Related Workmentioning

confidence: 99%

“…The first asymptotic approximation scheme (APTAS) for bin covering was introduced by Csirik et al [6]. Subsequently, Jansen and Solis-Oba [13] presented an AFPTAS for the problem. We discuss the relation between the two problems in Section 4.…”

Section: Related Workmentioning

confidence: 99%

“…Let A bc be such an AFPTAS, and let c be its asymptotic constant (such an AFPTAS with c = 4 is given in [13]). That is, given n items of sizes s(1), .…”

Section: Approximation Schemementioning

confidence: 99%

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Packing Resizable Items with Application to Video Delivery over Wireless Networks

Albagli-Kim

Epstein

Shachnai

et al. 2013

Algorithms for Sensor Systems

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Motivated by fundamental optimization problems in video delivery over wireless networks , we consider the following problem of packing resizable items (PRI). Given is a bin of capacity B > 0, and a set I of items. Each item j ∈ I is of size s j > 0. A packed item must stay in the bin for a fixed time interval. To accommodate more items in the bin, each item j can be compressed to a size p j ∈ [0, s j) for at most a fraction q j ∈ [0, 1) of the packing interval. The goal is to pack in the bin, for the given time interval, a subset of items of maximum cardinality. PRI is strongly NP-hard already for highly restricted instances. Our main result is an approximation algorithm that packs, for any instance I of PRI, at least 2 3 OP T (I) − 3 items, where OP T (I) is the number of items packed in an optimal solution. Our algorithm yields better ratio for instances in which the maximum compression time of an item is q max ∈ (0, 1 2). For subclasses of instances arising in realistic scenarios, we give an algorithm that packs at least OP T (I) − 2 items. Finally, we show that a non-trivial subclass of instances admits an asymptotic fully polynomial time approximation scheme (AFPTAS).

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Packing Resizable Items with Application to Video Delivery over Wireless Networks

Albagli-Kim

Epstein

Shachnai

et al. 2013

Algorithms for Sensor Systems

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“…This problem is the dual of the classical Bin Packing problem, where the goal is to pack the items into as few bins as possible; see the survey [2]. Bin Covering has received considerable attention in the past [1,[3][4][5][6]11]. We will survey relevant literature below.…”

Section: Models and Motivationmentioning

confidence: 99%

Approximation Algorithms for Generalized and Variable-Sized Bin Covering

Hellwig

Souza²

2012

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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In this paper, we consider the Generalized Bin Covering problem: We are given m bin types, where each bin of type i has profit p i and demand d i . Furthermore, there are n items, where item j has size s j . A bin of type i is covered if the set of items assigned to it has total size at least the demand d i . In that case, the profit of p i is earned and the objective is to maximize the total profit. To the best of our knowledge, only the cases p i = d i = 1 (Bin Covering) and p i = d i (Variable-Sized Bin Covering) have been treated before. We study two models of bin supply: In the unit supply model, we have exactly one bin of each type, i. e., we have individual bins. By contrast, in the infinite supply model, we have arbitrarily many bins of each type. Clearly, the unit supply model is a generalization of the infinite supply model, since we can simulate the latter with the former by introducing sufficiently many copies of each bin. To the best of our knowledge the unit supply model has not been studied yet. It is well-known that the problem in the infinite supply model is NP-hard, which can be seen by a straightforward reduction from Partition, and this hardness carries over to the unit supply model. This also implies that the problem can not be approximated better than two, unless P = NP.We begin our study with the unit supply model. Our results therein hold not only asymptotically, but for all instances. This contrasts most of the previous work on Bin Covering, which has been asymptotic. We prove that there is a combinatorial 5-approximation algorithm for Generalized Bin Covering with unit supply, which has running time O(nm √ m + n). This also transfers to the infinite supply model by the above argument. Furthermore, for Variable-Sized Bin Covering, in which we have p i = d i , we show that the natural and fast Next Fit Decreasing (nfd) algorithm is a 9 4-approximation in the unit supply model. The bound is tight for the algorithm and close to being best-possible, since the problem is inapproximable up to a factor of two, unless P = NP. Our analysis gives detailed insight into the limited extent to which the optimum can significantly outperform nfd.Then the question arises if we can improve on those results in asymptotic notions, where the optimal profit diverges. We discuss the difficulty of defining asymptotics in the unit supply model. For two natural definitions, the negative result holds that Variable-Sized Bin Covering in the unit supply model does not allow an APTAS. Clearly, this also excludes an APTAS for Generalized Bin Covering in that model. Nonetheless, we show that there is an AFPTAS for Variable-Sized Bin Covering in the infinite supply model.

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“…A straightforward reduction from the partition problem shows that it cannot be approximated within a factor of 1 /2 + ǫ for any ǫ > 0. On the positive side, Jansen and Solis-Oba [9] presented an asymptotic fully polynomial-time approximation scheme. In many applications, it is natural to study online variants, in which items arrive one after another and have to be packed directly into one of the bins without knowing the future items.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Analysis of the Dual Next-Fit Algorithm for Bin Covering

Fischer

Röglin

2016

LATIN 2016: Theoretical Informatics

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Abstract. In the bin covering problem, the goal is to fill as many bins as possible up to a certain minimal level with a given set of items of different sizes. Online variants, in which the items arrive one after another and have to be packed immediately on their arrival without knowledge about the future items, have been studied extensively in the literature. We study the simplest possible online algorithm Dual Next-Fit, which packs all arriving items into the same bin until it is filled and then proceeds with the next bin in the same manner. The competitive ratio of this and any other reasonable online algorithm is 1/2. We study Dual Next-Fit in a probabilistic setting where the item sizes are chosen i.i.d. according to a discrete distribution and we prove that, for every distribution, its expected competitive ratio is at least 1/2 + ǫ for a constant ǫ > 0 independent of the distribution. We also prove an upper bound of 2/3 and better lower bounds for certain restricted classes of distributions. Finally, we prove that the expected competitive ratio equals, for a large class of distributions, the random-order ratio, which is the expected competitive ratio when adversarially chosen items arrive in uniformly random order.

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An Asymptotic Fully Polynomial Time Approximation Scheme for Bin Covering

Cited by 12 publications

References 11 publications

Packing Resizable Items with Application to Video Delivery over Wireless Networks

Packing Resizable Items with Application to Video Delivery over Wireless Networks

Approximation Algorithms for Generalized and Variable-Sized Bin Covering

Probabilistic Analysis of the Dual Next-Fit Algorithm for Bin Covering

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