局内装箱算法综述

系统地介绍了局内装箱算法,归纳了其发展过程中的各种改进如数据分配模型、箱的划分等。阐述了该算法在工作分配、任务调度以及日常生活中的计划、包装、调度等计算机工程领域的应用。最后,对局内装箱算法提出了进一步的研究方向。     

Bin Packing with Rejection Revisited

We consider the following generalization of bin packing. Each item has a size in (0,1] associated with it, as well as a rejection cost, that an algorithm must pay if it chooses not to pack this item. The cost of an algorithm is the sum of all rejection costs of rejected items plus the number of unit sized bins used for packing all other items. We first study the offline version of the problem and design an APTAS for it. This is a non-trivial generalization of the APTAS given by Fernandez de la Vega and Lueker for the standard bin packing problem. We further give an approximation algorithm of an absolute approximation ratio 3/2, where this value is best possible unless P=NP.     

Selfish Bin Packing

Following recent interest in the study of computer science problems in a game theoretic setting, we consider the well known bin packing problem where the items are controlled by selfish agents. Each agent is charged with a cost according to the fraction of the used bin space its item requires. That is, the cost of the bin is split among the agents, proportionally to their sizes. Thus, the selfish agents prefer their items to be packed in a bin that is as full as possible. The social goal is to minimize the number of the bins used. The social cost in this case is therefore the number of bins used in the packing.     

Fair versus Unrestricted Bin Packing

Abstract. We consider the on-line Dual Bin Packing problem where we have n unit size bins and a sequence of items. The goal is to maximize the number of items that are packed in the bins by an on-line algorithm. We investigate unrestricted algorithms that have the power of performing admission control on the items, i.e., rejecting items while there is enough space to pack them, versus fair algorithms that reject an item only when there is not enough space to pack it. We show that by performing admission control on the items, we get better performance compared with the performance achieved on the fair version of the problem. Our main result shows that with an unfair variant of First-Fit, we can pack approximately two-thirds of the items for sequences for which an optimal off-line algorithm can pack all the items. This is in contrast to standard First-Fit where we show an asymptotically tight hardness result: if the number of bins can be chosen arbitrarily large, the fraction of the items packed by First-Fit comes arbitrarily close to five-eighths.     

多约束的箱子大小可变的装箱问题

作为经典装箱问题的扩展,带有约束条件的装箱问题有着很强的应用背景。从钢铁公司板坯入库引发的装箱问题中,物品除了重量这一参数,还引入了特性和种类等参数。将板坯入库归结为多约束的箱子大小可变的装箱问题后,建立了数学模型,将经典装箱算法进行了推广,构建了新的启发式算法A,它对输入物品按特性进行预处理,再进行装箱。理论分析了NFD,FFD,算法A的最坏情况的性能,并对这三种算法进行了数值模拟,得出在相关特性均匀分布下,算法A的性能最好。     

Approximation Algorithms for Extensible Bin Packing

In a variation of bin packing called extensible bin packing, the number of bins is specified as part of the input, and bins may be extended to hold more than the usual unit capacity. The cost of a bin is 1 if it is not extended, and the size if it is extended. The goal is to pack a set of items of given sizes into the specified number of bins so as to minimize the total cost. Adapting ideas Grötschel et al. (1981), Grötschel et al. (1988), Karmarkar and Karp (1982), Murgolo (1987), we give a fully polynomial time asymptotic approximation scheme (FPTAAS) for extensible bin packing. We close with comments on the complexity of obtaining stronger results.     

Online Square Packing with Gravity

We analyze the problem of packing squares in an online fashion: Given a semi-infinite strip of width 1 and an unknown sequence of squares of side length in [0,1] that arrive from above, one at a time. The objective is to pack these items as they arrive, minimizing the resulting height. Just like in the classical game of Tetris, each square must be moved along a collision-free path to its final destination. In addition, we account for gravity in both motion (squares must never move up) and position (any final destination must be supported from below). A similar problem has been considered before; the best previous result is by Azar and Epstein, who gave a 4-competitive algorithm in a setting without gravity (i.e., with the possibility of letting squares “hang in the air”) based on ideas of shelf packing: Squares are assigned to different horizontal levels, allowing an analysis that is reminiscent of some bin-packing arguments. We apply a geometric analysis to establish a competitive factor of 3.5 for the bottom-left heuristic and present a $\frac{34}{13} \approx 2.6154$ -competitive algorithm.     

受启动空间约束的装箱问题

提出了一种带有启动空间的约束装箱问题(start-up bin packing problem,简称SBPP),即不同类型的物品放入同一箱子中需要一个启动空间.该问题在工作分配、任务调度和日常生活中的包装等问题中有着广泛的应用背景.给出了一个求解SBPP的线性脱线算法C-NF,其最坏情况渐近性能比为2,与启动空间的大小无关.对该算法的平均性能进行了实验分析.另外,还分析了SBPP的在线特性,指出大量的经典在线装箱算法应用于SBPP都不存在确定的最坏情况渐近性能比,也给出了一种具有确定的最坏情况渐近性能比的在线算法.     

On Lazy Bin Covering and Packing problems

In this paper, we study two interesting variants of the classical bin packing problem, called Lazy Bin Covering (LBC) and Cardinality Constrained Maximum Resource Bin Packing (CCMRBP) problems. For the offline LBC problem, we first prove the approximation ratio of the First-Fit-Decreasing and First-Fit-Increasing algorithms, then present an APTAS. For the online LBC problem, we give a competitive analysis for the algorithms of Next-Fit, Worst-Fit, First-Fit, and a modified HARMONIC_M algorithm. The CCMRBP problem is a generalization of the Maximum Resource Bin Packing (MRBP) problem Boyar et al. (2006) [1]. For this problem, we prove that its offline version is no harder to approximate than the offline MRBP problem.     

两种准在线装箱算法

在装箱问题中，下次填充法（NF）由于其在线特性，而被广泛应用。然而这种算法由于按照物件到达先后顺序来填充，资源利用率比较低。该文根据计算机通信网络中的实际应用，在NF算法基础上添加了置换功能，提出两种新的算法：最后物件置换算法和每个物件置换算法。由于物件被置换后，可能会被填充到后续箱内，因此称之为准在线算法。通过平均分析发现，这两种算法性能比NF算法有较大提高，类似于智能NF算法的性能。     

Improved Approximation Algorithms for Maximum Resource Bin Packing and Lazy Bin Covering Problems

In this paper, we study two variants of the bin packing and covering problems called Maximum Resource Bin Packing (MRBP) and Lazy Bin Covering (LBC) problems, and present new approximation algorithms for them. For the offline MRBP problem, the previous best known approximation ratio is frac65frac{6}{5} (=1.2) achieved by the classical First-Fit-Increasing (FFI) algorithm (Boyar et al. in Theor. Comput. Sci. 362(1–3):127–139, 2006). In this paper, we give a new FFI-type algorithm with an approximation ratio of frac8071frac{80}{71} (≈1.12676). For the offline LBC problem, it has been shown in Lin et al. (COCOON, pp. 340–349, 2006) that the classical First-Fit-Decreasing (FFD) algorithm achieves an approximation ratio of frac7160frac{71}{60} (≈1.18333). In this paper, we present a new FFD-type algorithm with an approximation ratio of frac1715frac{17}{15} (≈1.13333). Our algorithms are based on a pattern-based technique and a number of other observations. They run in near linear time (i.e., O(nlog n)), and therefore are practical.     

带有一刀切约束的二维非规则装箱算法

针对切割下料领域的二维非规则一刀切装箱问题,首先给出了最小移动距离的定义,然后给出了一种基于最大移动距离的启发式算法。该算法通过计算一个凸多边形滑动至另一个凸多边形内部所允许的最大移动距离,对待排件的摆放位置进行一次性定位,避免使用传统的NFP(Not-Fit-Polygon)预判交方法,极大地缩短了排样的整体时间,最后使用模拟退火算法对下料流程进行了优化,改善了排样结果。     

An Improved Approximation Scheme for Variable-Sized Bin Packing

The Variable-Sized Bin Packing Problem (abbreviated as VSBPP or VBP) is a well-known generalization of the NP-hard Bin Packing Problem (BP) where the items can be packed in bins of M given sizes. The objective is to minimize the total capacity of the bins used. We present an asymptotic approximation scheme (AFPTAS) for VBP and BP with performance guarantee \(A_{\varepsilon }(I) \leq (1+ \varepsilon )OPT(I) + \mathcal {O}\left ({\log ^{2}\left (\frac {1}{\varepsilon }\right )}\right )\) for any problem instance I and any ε>0. The additive term is much smaller than the additive term of already known AFPTAS. The running time of the algorithm is \(\mathcal {O}\left ({ \frac {1}{\varepsilon ^{6}} \log \left ({\frac {1}{\varepsilon }}\right ) + \log \left ({\frac {1}{\varepsilon }}\right ) n}\right )\) for BP and \(\mathcal {O}\left ({ \frac {1}{{\varepsilon }^{6}} \log ^{2}\left ({\frac {1}{\varepsilon }}\right ) + M + \log \left ({\frac {1}{\varepsilon }}\right )n}\right )\) for VBP, which is an improvement to previously known algorithms. Many approximation algorithms have to solve subproblems, for example instances of the Knapsack Problem (KP) or one of its variants. These subproblems - like KP - are in many cases NP-hard. Our AFPTAS for VBP must in fact solve a generalization of KP, the Knapsack Problem with Inversely Proportional Profits (KPIP). In this problem, one of several knapsack sizes has to be chosen. At the same time, the item profits are inversely proportional to the chosen knapsack size so that the largest knapsack in general does not yield the largest profit. We introduce KPIP in this paper and develop an approximation scheme for KPIP by extending Lawler’s algorithm for KP. Thus, we are able to improve the running time of our AFPTAS for VBP.     

面向装箱问题的量子遗传优化算法

针对遗传算法系统的维持能力问题,提出一种量子演化算法(a Quantum-Inspired Evolutionary Algorithm)用于解决装箱问题的布局与优化。算法中采用量子比特编码、量子延伸变异操作。同时根据装箱问题具体情况,设计相应的量子旋转门更新策略,并在此基础上引入遗传操作,同时提出MCBF算法修复策略。最后,对8个测试数据集进行测试。实验测试结果显示,算法在维持遗传基因种群多样性与提高优化质量等方面效果明显。     

带约束的一维装箱问题近似算法的研究

作为经典装箱问题的扩展,有色装箱问题在多处理器实时调度的过程中有很强的应用背景。论文提出了有色装箱问题的新算法-SCPF算法,按颜色分类,将相同颜色的物品分成一类。放置时按照相同颜色的物品首先放置的原则,将物品进行装箱。实验证明,该算法与文献犤3犦中的KC-A算法相比具有更好的装箱效果,使用的箱子数更少。并从理论上论证了该算法的性能比KC-A算法更好。     

On Online Algorithms for Bin,Strip, and Box Packing,and Their Worst-Case and Average-Case Analysis

Programming and Computer Software - In this survey, we consider online algorithms for bin packing and strip packing problems, as well as their generalizations (multidimensional bin packing,...     

Online Removable Square Packing

The online removable square packing problem is a two-dimen-sional version of the online removable Knapsack problem. For a sequence of squares with side length at most 1, we are requested to pack a subset of them into a unit square bin in an online fashion where the online player can decide whether to take the current square or not and which squares currently in the unit square to remove. The goal is to maximize the total packed area. Our results include: (i) No online algorithm can achieve a better competitive ratio than . (ii) The matching upper bound is achieved by a relatively simple online algorithm if repacking is allowed. (iii) Without repacking, we can achieve an upper bound of 3 by using the concept of bricks of Januszewski and Lassak. (iv) The offline version of the problem admits a PTAS. Research supported by NSFC (10231060).