Eklavya Sharma
The harmonic function was first introduced by Lee and Lee (JACM 1985) for analyzing their online bin-packing algorithm. Subsequently, it has been used to obtain approximation algorithms for many different packing problems. Here we slightly generalize the harmonic function and give alternative proofs of its important properties.
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The 2D Online Bin Packing is a fundamental problem in Computer Science and the determination of its asymptotic competitive ratio has attracted great research attention. In a long series of papers, the lower bound of this ratio has been improved from 1.808, 1.856 to 1.907 and its upper bound reduced from 3.25, 3.0625, 2.8596, 2.7834 to 2.66013. In this paper, we rewrite the upper bound record to 2.5545. Our idea for the improvement is as follows. In SODA 2002 \cite{SS03}, Seiden and van Stee proposed an elegant algorithm called $H \otimes B$, comprised of the {\em Harmonic algorithm} $H$ and th…
We study the following variant of the classic {\em bin packing} problem. Given a set of items of various sizes, partitioned into groups, find a packing of the items in a minimum number of identical (unit-size) bins, such that no two items of the same group are assigned to the same bin. This problem, known as {\em bin packing with clique-graph conflicts}, has natural applications in storing file replicas, security in cloud computing and signal distribution. Our main result is an {\em asymptotic polynomial time approximation scheme (APTAS)} for the problem, improving upon the best known ratio of…
We study the 2-dimensional vector packing problem, which is a generalization of the classical bin packing problem where each item has 2 distinct weights and each bin has 2 corresponding capacities. The goal is to group items into minimum number of bins, without violating the bin capacity constraints. We propose a Θ}(n)-time approximation algorithm that is inspired by the O(n^2) algorithm proposed by Chang, Hwang, and Park.
We introduce the strongly NP-complete pagination problem, an extension of BIN PACKING where packing together two items may make them occupy less volume than the sum of their individual sizes. To achieve this property, an item is defined as a finite set of symbols from a given alphabet: while, in BIN PACKING, any two such sets would be disjoint, in PAGINATION, they can share zero, one or more symbols. After formulating the problem as an integer linear program, we try to approximate its solutions with several families of algorithms: from straightforward adaptations of classical BIN PACKING heuri…
In the online bin packing problem, a sequence of items is revealed one at a time, and each item must be packed into an available bin instantly upon its arrival. In this paper, we revisit the problem under a setting where the total number of items T is known in advance, also known as the closed online bin packing problem. Specifically, we study both the stochastic model and the random permutation model. We develop and analyze an adaptive algorithm that solves an offline bin packing problem at geometric time intervals and uses the offline optimal solution to guide online packing decisions. Under…
We explore approximation algorithms for the $d$-dimensional geometric bin packing problem ($d$BP). Caprara (MOR 2008) gave a harmonic-based algorithm for $d$BP having an asymptotic approximation ratio (AAR) of $T_{\infty}^{d-1}$ (where $T_{\infty} \approx 1.691$). However, their algorithm doesn't allow items to be rotated. This is in contrast to some common applications of $d$BP, like packing boxes into shipping containers. We give approximation algorithms for $d$BP when items can be orthogonally rotated about all or a subset of axes. We first give a fast and simple harmonic-based algorithm ha…
The Bin Packing Problem is one of the most important optimization problems. In recent years, due to its NP-hard nature, several approximation algorithms have been presented. It is proved that the best algorithm for the Bin Packing Problem has the approximation ratio 3/2 and the time order O(n), unless P=NP. In this paper, first, a 3/2-approximation algorithm is presented, then a modification to FFD algorithm is proposed to decrease time order to linear. Finally, these suggested approximation algorithms are compared with some other approximation algorithms. The experimental results show the sug…
In this paper we establish a general algorithmic framework between bin packing and strip packing, with which we achieve the same asymptotic bounds by applying bin packing algorithms to strip packing. More precisely we obtain the following results: (1) Any offline bin packing algorithm can be applied to strip packing maintaining the same asymptotic worst-case ratio. Thus using FFD (MFFD) as a subroutine, we get a practical (simple and fast) algorithm for strip packing with an upper bound 11/9 (71/60). A simple AFPTAS for strip packing immediately follows. (2) A class of Harmonic-based algorithm…
Packing-Inspired Algorithms for Periodic Scheduling Problems with Harmonic Periods
针对单机调和周期非抢占式调度问题，证明其与高度可分2D装箱的双射关系，提出约束规划建模及受装箱启发的首次适应启发式算法，在合成实例上展现出优越性能。
Since the Bin Packing Problem (BPP) is one of the main NP-hard problems, a lot of approximation algorithms have been suggested for it. It has been proven that the best algorithm for BPP has the approximation ratio of 3/2 and the time order of O(n), unless P=NP. In the current paper, a linear 3/2-approximation algorithm is presented. The suggested algorithm not only has the best possible theoretical factors, approximation ratio, space order, and time order, but also outperforms the other approximation algorithms according to the experimental results, therefore, we are able to draw the conclusio…
We continue the study of two recently introduced bin packing type problems, called bin packing with clustering, and online bin packing with delays. A bin packing input consists of items of sizes not larger than 1, and the goal is to partition or pack them into bins, where the total size of items of every valid bin cannot exceed 1. In bin packing with clustering, items also have colors associated with them. A globally optimal solution can combine items of different colors in bins, while a clustered solution can only pack monochromatic bins. The goal is to compare a globally optimal solution to …
Bin packing is a well studied problem involved in many applications. The classical bin packing problem is about minimising the number of bins and ignores how the bins are utilised. We focus in this paper, on a variant of bin packing that is at the heart of efficient management of data centres. In this context, servers can be viewed as bins and virtual machines as items. The efficient management of a data-centre involves minimising energy costs while ensuring service quality. The assignment of virtual machines on servers and how these servers are utilised has a huge impact on the energy consump…
We study a variant of online bin packing, called colorful bin packing. In this problem, items that are presented one by one are to be packed into bins of size 1. Each item i has a size s_i \in [0,1] and a color c_i \in C, where C is a set of colors (that is not necessarily known in advance). The total size of items packed into a bin cannot exceed its size, thus an item i can always be packed into a new bin, but an item cannot be packed into a non-empty bin if the previous item packed into that bin has the same color, or if the occupied space in it is larger than 1-s_i. This problem generalizes…
The bin-packing problem continues to remain relevant in numerous application areas. This technical report discusses the empirical performance of different bin-packing heuristics for certain test problems.
We present new approximation schemes for bin packing based on the following two approaches: (1) partitioning the given problem into mostly identical sub-problems of constant size and then construct a solution by combining the solutions of these constant size sub-problems obtained through PTAS or exact methods; (2) solving bin packing using irregular sized bins, a generalization of bin packing, that facilitates the design of simple and efficient recursive algorithms that solve a problem in terms of smaller sub-problems such that the unused space in bins used by an earlier solved sub-problem is …
The problem considered is the splittable bin packing with cardinality constraint. It is a variant of the bin packing problem where items are allowed to be split into parts but the number of parts in each bin is at most a given upper bound. Two versions of the splittable bin packing with cardinality constraint have been studied in the literature. Among these variants we consider the dual one where the objective is to minimize the maximum bin size while packing (may be fractional) the items to a given set of bins. We exhibit an EPTAS for the dual problem when the cardinality upper bound is part …
We consider the bin packing problem with $d$ different item sizes and revisit the structure theorem given by Goemans and Rothvoß[6] about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new structure theorem, we obtain an algorithm for the bin packing problem with running time $|V|^{2^{O(d)}} \cdot enc(I)^{O(1)}$, where $V$ is the set of vertices of the integer knapsack polytope and $enc(I)$ is the encoding length of the bin packin…
We present a new generalization of the extensible bin packing with unequal bin sizes problem. In our generalization the cost of exceeding the bin size depends on the index of the bin and not only on the amount in which the size of the bin is exceeded. This generalization does not satisfy the assumptions on the cost function that were used to present the existing polynomial time approximation scheme (PTAS) for the extensible bin packing with unequal bin sizes problem. In this work, we show the existence of an efficient PTAS (EPTAS) for this new generalization and thus in particular we improve t…
In the d-dimensional online bin packing problem, d-dimensional cubes of positive sizes no larger than 1 are presented one by one to be assigned to positions in d-dimensional unit cube bins. In this work, we provide improved upper bounds on the asymptotic competitive ratio for square and cube bin packing problems, where our bounds do not exceed 2.0885 and 2.5735 for square and cube packing, respectively. To achieve these results, we adapt and improve a previously designed harmonic-type algorithm, and apply a different method for defining weight functions. We detect deficiencies in the state-of-…
In this paper we present the first algorithm with optimal average-case and close-to-best known worst-case performance for the classic on-line problem of bin packing. It has long been observed that known bin packing algorithms with optimal average-case performance were not optimal in the worst-case sense. In particular First Fit and Best Fit had optimal average-case ratio of 1 but a worst-case competitive ratio of 1.7. The wasted space of First Fit and Best Fit for a uniform random sequence of length $n$ is expected to be $Θ(n^{2/3})$ and $Θ(\sqrt{n} \log ^{3/4} n)$, respectively. The competiti…
Bin Packing with Partition Matroid can be Approximated within $o(OPT)$ Bins
针对带划分拟阵约束的装箱问题，提出基于配置-LP舍入的多项式时间算法，实现$OPT+o(OPT)$个箱子的近似，改进并推广了现有结果。
We consider two well-known natural variants of bin packing, and show that these packing problems admit asymptotic fully polynomial time approximation schemes (AFPTAS). In bin packing problems, a set of one-dimensional items of size at most 1 is to be assigned (packed) to subsets of sum at most 1 (bins). It has been known for a while that the most basic problem admits an AFPTAS. In this paper, we develop methods that allow to extend this result to other variants of bin packing. Specifically, the problems which we study in this paper, for which we design asymptotic fully polynomial time approxim…
We consider several previously studied online variants of bin packing and prove new and improved lower bounds on the asymptotic competitive ratios for them. For that, we use a method of fully adaptive constructions. In particular, we improve the lower bound for the asymptotic competitive ratio of online square packing significantly, raising it from roughly 1.68 to above 1.75.
Following the work of Anily et al., we consider a variant of bin packing, called "bin packing with general cost structures" (GCBP) and design an asymptotic fully polynomial time approximation scheme (AFPTAS) for this problem. In the classic bin packing problem, a set of one-dimensional items is to be assigned to subsets of total size at most 1, that is, to be packed into unit sized bins. However, in GCBP, the cost of a bin is not 1 as in classic bin packing, but it is a non-decreasing and concave function of the number of items packed in it, where the cost of an empty bin is zero. The construc…
The Sum of Squares algorithm for bin packing was defined in [2] and studied in great detail in [1], where it was proved that its worst case performance ratio is at most 3. In this note, we improve the asymptotic worst case bound to 2.7777...
We consider a known variant of bin packing called {\it cardinality constrained bin packing}, also called {\it bin packing with cardinality constraints} (BPCC). In this problem, there is a parameter k\geq 2, and items of rational sizes in [0,1] are to be packed into bins, such that no bin has more than k items or total size larger than 1. The goal is to minimize the number of bins. A recently introduced concept, called the price of clustering, deals with inputs that are presented in a way that they are split into clusters. Thus, an item has two attributes which are its size and its cluster. The…
Bin Packing with Minimum Color Fragmentation (BPMCF) is an extension of the Bin Packing Problem in which each item has a size and a color and the goal is to minimize the sum of the number of bins containing items of each color. In this work, we introduce BPMCF and present a decomposition strategy to solve the problem, where the assignment of items to bins is formulated as a binary decision diagram and an optimal integrated solutions is identified through a mixed-integer linear programming model. Our computational experiments show that the proposed approach greatly outperforms a direct formulat…
In this paper we propose an improved approximation scheme for the Vector Bin Packing problem (VBP), based on the combination of (near-)optimal solution of the Linear Programming (LP) relaxation and a greedy (modified first-fit) heuristic. The Vector Bin Packing problem of higher dimension (d \geq 2) is not known to have asymptotic polynomial-time approximation schemes (unless P = NP). Our algorithm improves over the previously-known guarantee of (ln d + 1 + epsilon) by Bansal et al. [1] for higher dimensions (d > 2). We provide a θ(1) approximation scheme for certain set of inputs for any dime…
APTAS for bin packing with general cost structures
针对带一般基数依赖成本的推广装箱问题，提出其APTAS，并给出该问题随成本函数f变化的完整复杂度分类。
We improve the lower bound on the asymptotic competitive ratio of any online algorithm for bin packing to above 1.54278. We demonstrate for the first time the advantage of branching and the applicability of full adaptivity in the design of lower bounds for the classic online bin packing problem. We apply a new method for weight based analysis, which is usually applied only in proofs of upper bounds. The values of previous lower bounds were approximately 1.5401 and 1.5403.
We study the Colored Bin Packing Problem: we are given a set of items where each item has a weight and color. We must pack the items in bins of uniform capacity such that no two items of the same color may be adjacent within in a bin. The goal is to perform this packing using the fewest number of bins. We consider a version of the problem where reordering is allowed. We first consider the zero-weight and unit weight versions of this problem, i.e. where the items have weight zero and one, respectively. We present linear time optimal algorithms for both versions.
Offline green bin packing and its constrained variant
本文提出了一种APTAS和1.5近似算法，解决绿色装箱及其约束变体问题，使bin数量和能耗总和最小化。
We analyze a recently introduced concept, called the price of clustering, for variants of bin packing called open-end bin packing problems (OEBP). Input items have sizes, and they also belong to a certain number of types. The new concept deals with the comparison of optimal solutions for the cases where items of distinct types can and cannot be packed together, respectively. The problem is related to greedy bin packing algorithms and to batched bin packing, and we discuss some of those concepts as well. We analyze max-OEBP, where a packed bin is valid if by excluding its largest item, the tota…
Green Bin Packing
本文提出了绿色装箱问题，一种具有超过固定水平线性成本的在线装箱变体，并针对不同参数条件设计了改进的算法变体，在理论与实验上均验证了其性能优势。
In the online bin packing problem, items of sizes in (0,1] arrive online to be packed into bins of size 1. The goal is to minimize the number of used bins. In this paper, we present an online bin packing algorithm with asymptotic competitive ratio of 1.5813. This is the first improvement in fifteen years and reduces the gap to the lower bound by 15%. Within the well-known SuperHarmonic framework, no competitive ratio below 1.58333 can be achieved. We make two crucial changes to that framework. First, some of our algorithm's decisions depend on exact sizes of items, instead of only their types.…
We consider a variant of the classical Bin Packing Problem, called Fully Dynamic Bin Packing. In this variant, items of a size in $(0,1]$ must be packed in bins of unit size. In each time step, an item either arrives or departs from the packing. An algorithm for this problem must maintain a feasible packing while only repacking a bounded number of items in each time step. We develop an algorithm which repacks only a constant number of items per time step and, unlike previous work, does not rely on bundling of small items which allowed those solutions to move an unbounded number of small items …
The vector bin packing problem (VBP) is a generalization of bin packing with multiple constraints. In this problem we are required to pack items, represented by p-dimensional vectors, into as few bins as possible. The multiple-choice vector bin packing (MVBP) is a variant of the VBP in which bins have several types and items have several incarnations. We present an exact method, based on an arc-flow formulation with graph compression, for solving MVBP by simply representing all the patterns in a very compact graph. As a proof of concept we report computational results on a variable-sized bin p…
Cardinality constrained bin packing or bin packing with cardinality constraints is a basic bin packing problem. In the online version with the parameter k \geq 2, items having sizes in (0,1] associated with them are presented one by one to be packed into unit capacity bins, such that the capacities of bins are not exceeded, and no bin receives more than k items. We resolve the online problem in the sense that we prove a lower bound of 2 on the overall asymptotic competitive ratio. This closes this long standing open problem, since an algorithm of an absolute competitive ratio 2 is known. Addit…
Bin packing with cardinality constraints is a bin packing problem where an upper bound k \geq 2 on the number of items packed into each bin is given, in addition to the standard constraint on the total size of items packed into a bin. We study the online scenario where items are presented one by one. We analyze it with respect to the absolute competitive ratio and prove tight bounds of 2 for any k \geq 4. We show that First Fit also has an absolute competitive ratio of 2 for k=4, but not for larger values of k, and we present a complete analysis of its asymptotic competitive ratio for all valu…
We consider several extensions of the fractional bin packing problem, a relaxation of the traditional bin packing problem where the objects may be split across multiple bins. In these extensions, we introduce load-balancing constraints imposing that the share of each object which is assigned to a same bin must be equal. We propose a Mixed-Integer Programming (MIP) formulation and show that the problem becomes NP-hard if we limit to at most 3 the number of bins across which each object can be split. We then consider a variant where the balanced allocations of objects to bins may be done in succ…
We consider the problem of packing rectangles into bins that are unit squares, where the goal is to minimize the number of bins used. All rectangles have to be packed non-overlapping and orthogonal, i.e., axis-parallel. We present an algorithm for this problem with an absolute worst-case ratio of 2, which is optimal provided P != NP.
In this paper, we study online multidimensional bin packing problem when all items are hypercubes. Based on the techniques in one dimensional bin packing algorithm Super Harmonic by Seiden, we give a framework for online hypercube packing problem and obtain new upper bounds of asymptotic competitive ratios. For square packing, we get an upper bound of 2.1439, which is better than 2.24437. For cube packing, we also give a new upper bound 2.6852 which is better than 2.9421 by Epstein and van Stee.
On data reduction for dynamic vector bin packing
该研究针对动态向量装箱问题，证明了相关数据约简的难度，提出一种可获取任意ε>0下(1+ε)近似解的多项式时间约简算法，在ε=0.02时可将两大云平台实例规模缩减一个数量级。
We consider a variant of bin packing called multiple-choice vector bin packing. In this problem we are given a set of items, where each item can be selected in one of several $D$-dimensional incarnations. We are also given $T$ bin types, each with its own cost and $D$-dimensional size. Our goal is to pack the items in a set of bins of minimum overall cost. The problem is motivated by scheduling in networks with guaranteed quality of service (QoS), but due to its general formulation it has many other applications as well. We present an approximation algorithm that is guaranteed to produce a sol…
In the Colored Bin Packing problem a set of items with varying weights and colors must be packed into bins of uniform weight limit such that no two items of the same color may be packed adjacently within a bin. We solve this problem for the case where there are two or more colors when the items have zero weight and when the items have unit weight. Our algorithms are optimal and run in linear time. Since our algorithms apply for two or more colors, they demonstrate that the problem does not get harder as the number of colors increases. We also provide closed-form expressions for the optimal num…
Recently, Renault (2016) studied the dual bin packing problem in the per-request advice model of online algorithms. He showed that given $O(1/ε)$ advice bits for each input item allows approximating the dual bin packing problem online to within a factor of $1+ε$. Renault asked about the advice complexity of dual bin packing in the tape-advice model of online algorithms. We make progress on this question. Let $s$ be the maximum bit size of an input item weight. We present a conceptually simple online algorithm that with total advice $O\left(\frac{s + \log n}{ε^2}\right)$ approximates the dual b…
Column-generation for a two-dimensional multi-criteria bin-packing problem
针对印刷电路板制造中需同时最小化箱数与布局种类的二维多目标装箱问题，提出带改进Ryan-Foster分支的分支定价列生成方法，分离耗时约束以提升求解效率。
$k$-times bin packing and its application to fair electricity distribution
该研究提出了$k$次装箱新变体，推广经典装箱近似算法并分析性能比，将其应用于发展中国家公平电力分配问题，验证了其在连接时间平等分配上优于现有启发式方案。
We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providin…
On the 2D Demand Bin Packing Problem: Hardness and Approximation Algorithms
本文研究二维需求装箱问题，证明短高度任务和正方形任务变体在优于2的因子内近似是NP困难的，并基于通用框架结合First-Fit推广给出最优近似算法及一般情形的3近似算法。
For bin packing, the input consists of n items with sizes s_1,...,s_n in [0,1] which have to be assigned to a minimum number of bins of size 1. The seminal Karmarkar-Karp algorithm from '82 produces a solution with at most OPT + O(log^2 OPT) bins. We provide the first improvement in now 3 decades and show that one can find a solution of cost OPT + O(log OPT * log log OPT) in polynomial time. This is achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation using the Entropy Method from discrepancy theory. The result is constructive via algorithms of Bansal and Lovett-Meka.
In this paper we present a fast scalable heuristic for bin packing that partitions the given problem into identical sub-problems of constant size and solves these constant size sub-problems by considering only a constant number of bin configurations with bounded unused space. We present some empirical evidence to support the scalability of our heuristic and its tighter empirical analysis of hard instances due to improved lower bound on the necessary wastage in an optimal solution.
Packing a given sequence of items into as few bins as possible in an online fashion is a widely studied problem. We improve lower bounds for packing boxes into bins in two or more dimensions, both for general algorithms for squares and rectangles (in two dimensions) and for an important subclass, so-called Harmonic-type algorithms for hypercubes (in two or more dimensions). Lastly, we show that two adaptions of ideas from a one-dimensional packing algorithm to square packing do not help to break the barrier of 2.
Online Bin Packing with Item Size Estimates
本文提出带尺寸估计的在线装箱模型，给出4/3下界、δ≤1/35时的1.5竞争算法，以及每箱至多两件时最优的4/3策略。
An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no "small" items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number $k$ of small items. Our main results are fixed…
We consider the Ordered Open End Bin Packing problem. Items of sizes in $(0,1]$ are presented one by one, to be assigned to bins in this order. An item can be assigned to any bin for which the current total size strictly below $1$. This means also that the bin can be overloaded by its last packed item. We improve lower and upper bounds on the asymptotic competitive ratio in the online case. Specifically, we design the first algorithm whose asymptotic competitive ratio is strictly below $2$ and it is close to the lower bound. This is in contrast to the best possible absolute approximation ratio…
In a recent paper, Brusco, Köhn and Steinley [Ann. Oper. Res. 206:611-626 (2013)] conjecture that the 2 bins special case of the one-dimensional minimax bin-packing problem with bin size constraints might be solvable in polynomial time. In this note, we show that this problem is NP-hard for the special case and that it is strongly NP-hard for the general problem. We also propose a pseudo-polynomial algorithm for the special case and a constructive heuristic for the general problem.
Improved Hardness of Approximation for Geometric Bin Packing
本文针对几何装箱问题，证明除非NP=P，否则该问题无法在d^{1-ε}因子内近似，并引入集合族的几何装箱维度概念，证明其关键性质以揭示几何装箱与向量装箱的根本差异。
We solve the Bin Packing problem in $O^*(2^k)$ time, where $k$ is the number of items less or equal to one third of the bin capacity. This parameter measures the distance from the polynomially solvable case of only large (i.e., greater than one third) items. Our algorithm is actually designed to work for a more general Vector Bin Packing problem, in which items are multidimensional vectors. We improve over the previous fastest $O^*(k! \cdot 4^k)$ time algorithm. Our algorithm works by reducing the problem to finding an exact weight perfect matching in a (multi-)graph with $O^*(2^k)$ edges, who…
We consider the online vector bin packing problem where $n$ items specified by $d$-dimensional vectors must be packed in the fewest number of identical $d$-dimensional bins. Azar et al. (STOC'13) showed that for any online algorithm $A$, there exist instances I, such that $A(I)$, the number of bins used by $A$ to pack $I$, is $Ω(d/\log^2 d)$ times $OPT(I)$, the minimal number of bins to pack $I$. However in those instances, $OPT(I)$ was only $O(\log d)$, which left open the possibility of improved algorithms with better asymptotic competitive ratio when $OPT(I) \gg d$. We rule this out by show…
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