Online Bin Packing Research Articles

Tight bounds for NF-based bounded-space online bin packing algorithms

In Zheng et al. (J Comb Optim 30(2):360–369, 2015) modelled a surgery problem by the one-dimensional bin packing, and developed a semi-online algorithm to give an efficient feasible solution. In their algorithm they used a buffer to temporarily store items, having a possibility to lookahead in the list. Because of the considered practical problem they investigated the 2-parametric case, when the size of the items is at most 1 / 2. Using an NF-based online algorithm the authors proved an ACR of $$13/9 = 1.44\ldots $$ for any given buffer size not less than 1. They also gave a lower bound of $$4/3=1.33\ldots $$ for the bounded-space algorithms that use NF-based rules. Later, in Zhang et al. (J Comb Optim 33(2):530–542, 2017) an algorithm was given with an ACR of 1.4243, and the authors improved the lower bound to 1.4230. In this paper we present a tight lower bound of $$h_\infty (r)$$ for the r-parametric problem when the buffer capacity is 3. Since $$h_\infty (2) = 1.42312\ldots $$ , our result—as a special case—gives a tight bound for the algorithm-class given in 2017. To prove that the lower bound is tight, we present an NF-based online algorithm that considers the r-parametric problem, and uses a buffer with capacity of 3. We prove that this algorithm has an ACR that is equal to the lower bounds for arbitrary r.

CHAMP: Creating heuristics via many parameters for online bin packing

The online bin packing problem is a well-known bin packing variant and which requires immediate decisions to be made for the placement of a lengthy sequence of arriving items of various sizes one at a time into fixed capacity bins without any overflow. The overall goal is maximising the average bin fullness. We investigate a ‘policy matrix’ representation, which assigns a score for each decision option independently and the option with the highest value is chosen, for one-dimensional online bin packing. A policy matrix might also be considered as a heuristic with many parameters, where each parameter value is a score. We hence effectively investigate a framework which can be used for creating heuristics via many parameters. The proposed framework combines a Genetic Algorithm optimiser, which searches the space of heuristics in policy matrix form, and an online bin packing simulator, which acts as the evaluation function. The empirical results indicate the success of the proposed approach, providing the best solutions for almost all item sequence generators used during the experiments. We also present a novel fitness landscape analysis on the search space of policies. This study hence gives evidence of the potential for automated discovery by intelligent systems of powerful heuristics for online problems; reducing the need for expensive use of human expertise.

Bounds for online bin packing with cardinality constraints

We study a bin packing problem in which a bin can contain at most k items of total size at most 1, where k≥2 is a given parameter. Items are presented one by one in an online fashion. We analyze the best absolute competitive ratio of the problem and prove tight bounds of 2 for any k≥4. Additionally, we present bounds for relatively small values of k with respect to the asymptotic competitive ratio and the absolute competitive ratio. In particular, we provide tight bounds on the absolute competitive ratio of First Fit for k=2,3,4, and improve the known lower bounds on asymptotic competitive ratios for multiple values of k. Our method for obtaining a lower bound on the asymptotic competitive ratio using a certain type of an input is general, and we also use it to obtain an alternative proof of the known lower bound on the asymptotic competitive ratio of standard online bin packing.

Lower bound for 3-batched bin packing

In this paper we will consider a special relaxation of the well-known online bin packing problem. In a batched bin packing problem (BBPP)–defined by Gutin et al. (2005)–the elements come in batches and one batch is available for packing in a given time. If we have K≥2 batches then we denote the problem by K-BBPP. In Gutin et al. (2005) the authors gave a 1.3871… lower bound for the asymptotic competitive ratio (ACR) of any on-line 2-BBBP algorithm. In this paper we investigate the 3-BBPP, and we give 1.51211… lower bound for its ACR.

Online bin packing problem with buffer and bounded size revisited

In this paper we study the online bin packing with buffer and bounded size, i.e., there are items with size within $$(\alpha ,1/2]$$(ź,1/2] where $$0 \le \alpha < 1/2 $$0≤ź<1/2, and there is a buffer with constant size. Each time when a new item is given, it can be stored in the buffer temporarily or packed into some bin, once it is packed in the bin, we cannot repack it. If the input is ended, the items in the buffer should be packed into bins too. In our setting, any time there is at most one bin open, i.e., that bin can accept new items, and all the other bins are closed. This problem is first studied by Zheng et al. (J Combin Optim 30(2):360---369, 2015), and they proposed a 1.4444-competitive algorithm and a lower bound 1.3333 on the competitive ratio. We improve the lower bound from 1.3333 to 1.4230, and the upper bound from 1.4444 to 1.4243.

Online algorithms for 2D bin packing with advice

In advice complexity model of online bin packing, suppose there is an offline oracle with infinite computational power. The oracle can provide some information to the online calculation about future items after scanning the whole list of items. This approach can loosen online constraints. The online algorithm receives an advice bit upon the arrival of each item and makes the packing decision. In 2-dimensional online bin packing with advice problem, each item is a rectangle of side lengths less than or equal to 1. The items are to be packed into square bins of size 1×1, without overlapping, allowing 90° rotations. The goal is to pack the items into the minimum number of square bins. In this paper, a 2.25-competitive 2-dimensional online rectangular packing algorithm with three bits of advice per item is presented. Furthermore, an online strategy with competitive ratio 2 requiring three bits of advice per item is given for square packing.

An asymptotic competitive scheme for online bin packing

In the online bin packing problem, a list of items with integral sizes between 1 to B arrive one by one. Each item must be irrevocably assigned into a bin of capacity B upon its arrival without any information on the subsequent items, and the goal is to minimize the number of used bins. In this paper, we deal with online bin packing from a new sight. We present an asymptotic competitive scheme, i.e., for any ϵ>0, the asymptotic competitive ratio is at most ρ⁎+ϵ, where ρ⁎ is the smallest possible asymptotic competitive ratio among all online algorithms. Apart from the technical results, the analysis to bridge the online and the off-line approaches might be of particular interests.

SIGACT News Online Algorithms Column 26

I mentioned in a previous column [22] that the best known lower bound for the two-dimensional online bin packing problem is 1.907 by Blitz, van Vliet, Woeginger [2], which is an unpublished (and now lost [24]) manuscript. I have realized since then that even the penultimate result, 1.856, was published only in Andre van Vliet’s Ph.D. thesis [23] and is not readily available. It therefore seems like a good idea to describe his methods here, though not in full detail. This will include a discussion of the three-dimensional case. I will also survey other results in multidimensional packing, that were left out in my previous column. I would like to invite more contributions to this column, be it surveys, conference reports, or technical articles related to online algorithms and competitive analysis. If you are considering becoming a guest writer, don’t hesitate to mail me at rvs4@le.ac.uk.

Online Bin Packing with Advice

We consider the online bin packing problem under the advice complexity model where the “online constraint” is relaxed and an algorithm receives partial information about the future items. We provide tight upper and lower bounds for the amount of advice an algorithm needs to achieve an optimal packing. We also introduce an algorithm that, when provided with \(\log n + o(\log n)\) bits of advice, achieves a competitive ratio of \(3/2\) for the general problem. This algorithm is simple and is expected to find real-world applications. We introduce another algorithm that receives \(2n + o(n)\) bits of advice and achieves a competitive ratio of \(4/3 + \varepsilon \). Finally, we provide a lower bound argument that implies that advice of linear size is required for an algorithm to achieve a competitive ratio better than 9/8.

Adaptive Resource Provisioning for the Cloud Using Online Bin Packing

Data center applications present significant opportunities for multiplexing server resources. Virtualization technology makes it easy to move running application across physical machines. In this paper, we present an approach that uses virtualization technology to allocate data center resources dynamically based on application demands and support green computing by optimizing the number of servers actively used. We abstract this as a variant of the relaxed on-line bin packing problem and develop a practical, efficient algorithm that works well in a real system. We adjust the resources available to each VM both within and across physical servers. Extensive simulation and experiment results demonstrate that our system achieves good performance compared to the existing work.

NF-based algorithms for online bin packing with buffer and bounded item size

This paper studies a variation of online bin packing where there is a capacitated buffer to temporarily store items during packing, and item size is bounded within $$(\alpha , 1/2]$$(?,1/2] for some $$0\le \alpha < 1/2$$0≤?<1/2. The problem is motivated by surgery scheduling such that we regard the planned uniform available time interval in each day as a unit size bin and surgeries as items to be packed. Our focus is on the asymptotic performance of NF (Next Fit) and NF-based online algorithms. We show that the classical NF algorithm without use of the buffer has an asymptotic competitive ratio of $$2/(1+\alpha )$$2/(1+?). An NF-based algorithm which makes use of the buffer is further proposed, and proved to be asymptotic 13/9-competitive for any given buffer size not less than 1. We also present a lower bound of 4/3.

Online bin covering: Expectations vs. guarantees

Bin covering is a dual version of classic bin packing. Thus, the goal is to cover as many bins as possible, where covering a bin means packing items of total size at least one in the bin.For online bin covering, competitive analysis fails to distinguish between most algorithms of interest; all “reasonable” algorithms have a competitive ratio of 12. Thus, in order to get a better understanding of the combinatorial difficulties in solving this problem, we turn to other performance measures, namely relative worst order, random order, and max/max analysis, as well as analyzing input with restricted or uniformly distributed item sizes. In this way, our study also supplements the ongoing systematic studies of the relative strengths of various performance measures.Two classic algorithms for online bin packing that have natural dual versions are Harmonick and Next-Fit. Even though the algorithms are quite different in nature, the dual versions are not separated by competitive analysis. We make the case that when guarantees are needed, even under restricted input sequences, dual Harmonick is preferable. In addition, we establish quite robust theoretical results showing that if items come from a uniform distribution or even if just the ordering of items is uniformly random, then dual Next-Fit is the right choice.

Online bin packing with (1,1) and (2, $$R$$ R ) bins

We study a variant of online bin packing problem, in which there are two types of bins: $$(1,1)$$(1,1) and $$(2,R)$$(2,R), i.e., unit size bin with cost 1 and size 2 bin with cost $$R > 1$$R>1, the objective is to minimize the total cost occurred when all the items are packed into the two types of bins. When $$R > 3$$R>3, the offline version of this problem is equivalent to the classical bin packing problem. In this paper, we focus on the case $$ R \le 3$$R≤3, and propose online algorithms and obtain lower bounds for the problem.

Online Results for Black and White Bin Packing

In online bin packing problems, items of sizes in [0, 1] are to be partitioned into subsets of total size at most 1, called bins. We introduce a new variant where items are of two types, called black and white, and the item types must alternate in each bin, that is, two items of the same type cannot be assigned consecutively into a bin. This variant generalizes the standard online bin packing problem. We design an online algorithm with an absolute competitive ratio of 3. We further show that a number of well-known algorithms cannot have a better performance, even in the asymptotic sense. Interestingly, we show that this problem is harder than standard online bin packing by proving a general lower bound 1 + 1 2 ln 2 ? 1 . 7213 $1+\frac {1}{2\ln 2}\approx 1.7213$ on the asymptotic competitive ratio of any deterministic or randomized online algorithm. This lower bound exceeds the upper bounds known for the absolute competitive ratio of standard online bin packing.

Applying Gene Expression Programming for Solving One-Dimensional Bin-Packing Problems

This work aims to study and explore the use of Gene Expression Programming (GEP) in solving on-line Bin-Packing problem. The main idea is to show how GEP can automatically find acceptable heuristic rules to solve the problem efficiently and economically. One dimensional Bin-Packing problem is considered in the course of this work with the constraint of minimizing the number of bins filled with the given pieces. Experimental Data includes instances of benchmark test data taken from Falkenauer (1996) for One-dimensional Bin-Packing Problems. Results show that GEP can be used as a very powerful and flexible tool for finding interesting compact rules suited for the problem. The impact of functions is also investigated to show how they can affect and influence the success of rates when they appear in rules. High success rates are gained with smaller population size and fewer generations compared to a previous work performed using Genetic Programming.

An Algorithm for Online Two-dimensional Bin Packing with Two Open Bins

Bin packing problem has been used for many applications. Its bounded space variant has more practical value because of the limited resources in the real world. In this paper, two-dimensional online bin packing problem with 2-bounded space is considered: A sequence of two-dimensional rectangles is asked to be packed into a minimum number of unit square bins. The packing is in an on-line manner that each rectangle item should be packed immediately upon its arrival. No two items overlap. Only two bins are available at any time. We extend the idea of one-bounded space rectangle packing by Zhang et al. [1] to 2-bounded space, modify the classification of items, pack items by using different strategies into two open bins and obtain a 4:4138-competitive ratio. Comparing to Zhang’s result of 5:155, this method is actually good when open two bins during the packing processing.

Improved lower bounds for the online bin packing problem with cardinality constraints

The bin packing problem has been extensively studied and numerous variants have been considered. The $$k$$ k -item bin packing problem is one of the variants introduced by Krause et al. (J ACM 22:522---550, 1975). In addition to the formulation of the classical bin packing problem, this problem imposes a cardinality constraint that the number of items packed into each bin must be at most $$k$$ k . For the online setting of this problem, in which the items are given one by one, Babel et al. (Discret Appl Math 143:238---251, 2004) provided lower bounds $$\sqrt{2} \approx 1.41421$$ 2 ? 1.41421 and $$1.5$$ 1.5 on the asymptotic competitive ratio for $$k=2$$ k = 2 and $$3$$ 3 , respectively. For $$k \ge 4$$ k ? 4 , some lower bounds (e.g., by van Vliet (Inf Process Lett 43:277---284, 1992) for the online bin packing problem, i.e., a problem without cardinality constraints, can be applied to this problem. In this paper we consider the online $$k$$ k -item bin packing problem. First, we improve the previous lower bound $$1.41421$$ 1.41421 to $$1.42764$$ 1.42764 for $$k=2$$ k = 2 . Moreover, we propose a new method to derive lower bounds for general $$k$$ k and present improved bounds for various cases of $$k \ge 4$$ k ? 4 . For example, we improve $$1.33333$$ 1.33333 to $$1.5$$ 1.5 for $$k = 4$$ k = 4 , and $$1.33333$$ 1.33333 to $$1.47058$$ 1.47058 for $$k = 5$$ k = 5 .

Online bin packing with delay and holding costs

We consider online bin packing where in addition to the opening cost of each bin, the arriving items collect delay costs until their assigned bins are released (closed), and the open bins themselves collect holding costs. Besides being of practical interest, this problem generalises several previously unrelated online optimisation problems. We provide a general online algorithm for this problem with competitive ratio 7, with improvements for the special cases of zero delay or holding costs and size-proportional item delay costs.

Semi-on-line bin packing: a short overview and a new lower bound

Here we review the main results in the area of semi-on-line bin packing. Then we present a new lower bound for the asymptotic competitive ratio of any on-line bin packing algorithm which knows the optimum value in advance.

On-line algorithms for 2-space bounded 2-dimensional bin packing

In 2-space bounded model of on-line bin packing, there are 2 active bins, and each item can be packed only into one of the active bins. If it is impossible to pack an item into any active bin, we close one of the current active bins and open a new active bin to pack that item. In 2-space bounded 2-dimensional bin packing problem, each item is a rectangle of side lengths not greater than 1. The items are packed on-line into square bins of size 1×1 and 90°-rotations are allowed. In this paper a 4-competitive 2-space bounded 2-dimensional on-line packing algorithm is presented. Furthermore, an on-line strategy with competitive ratio 3.8 is given for 2-space bounded square packing.