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 dimension d. More precisely, we provide a 2-OPT algorithm, a result which is irrespective of the number of dimensions d.
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 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 …
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 …
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-…
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…
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.
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.
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…
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 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…
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…
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.
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.
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.
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…
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.
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…
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.
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…
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 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…
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…
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 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…
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…
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 revisit the classic online bin packing problem. In this problem, items of positive sizes no larger than 1 are presented one by one to be packed into subsets called "bins" of total sizes no larger than 1, such that every item is assigned to a bin before the next item is presented. We use online partitioning of items into classes based on sizes, as in previous work, but we also apply a new method where items of one class can be packed into more than two types of bins, where a bin type is defined according to the number of such items grouped together. Additionally, we allow the smallest class …
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…
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…
In this work, we consider online vector bin packing. It is known that no algorithm can have a competitive ratio of $o(d/\log^2 d)$ in the absolute sense, though upper bounds for this problem were always shown in the asymptotic sense. Since variants of bin packing are traditionally studied with respect to the asymptotic measure and since the two measures are different, we focus on the asymptotic measure and prove new lower bounds on the asymptotic competitive ratio. The existing lower bounds prior to this work were much smaller than $3$ even for very large dimensions. We significantly improve t…
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...
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…
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…
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.
We study the well-known two-dimensional strip packing problem. Given is a set of rectangular axis-parallel items and a strip of width $W$ with infinite height. The objective is to find a packing of these items into the strip, which minimizes the packing height. Lately, it has been shown that the lower bound of $3/2$ of the absolute approximation ratio can be beaten when we allow a pseudo-polynomial running-time of type $(n W)^{f(1/\varepsilon)}$. If $W$ is polynomially bounded by the number of items, this is a polynomial running-time. We present a pseudo-polynomial algorithm with approximation…
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 …
We slightly improve the known lower bound on the asymptotic competitive ratio for online bin packing of rectangles. We present a complete proof for the new lower bound, whose value is above 1.91.
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.
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 Conflicts (BPC) are problems in which items with compatibility constraints must be packed in the least number of bins, not exceeding the capacity of the bins and ensuring that non-conflicting items are packed in each bin. In this work, we introduce the Bin Packing Problem with Compatible Categories (BPCC), a variant of the BPC in which items belong to conflicting or compatible categories, in opposition to the item-by-item incompatibility found in previous literature. It is a common problem in the context of last mile distribution to nanostores located in densely populated area…
The Two-dimensional Bin Packing Problem calls for packing a set of rectangular items into a minimal set of larger rectangular bins. Items must be packed with their edges parallel to the borders of the bins, cannot be rotated and cannot overlap among them. The problem is of interest because it models many real-world applications, including production, warehouse management and transportation. It is, unfortunately, very difficult, and instances with just 40 items are unsolved to proven optimality, despite many attempts, since the 1990s. In this paper, we solve the problem with a combinatorial Ben…
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 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…
Motivated by bursty bandwidth allocation and by the allocation of virtual machines to servers in the cloud, we consider the online problem of packing items with random sizes into unit-capacity bins. Items arrive sequentially, but upon arrival an item's actual size is unknown; only its probabilistic information is available to the decision maker. Without knowing this size, the decision maker must irrevocably pack the item into an available bin or place it in a new bin. Once packed in a bin, the decision maker observes the item's actual size, and overflowing the bin is a possibility. An overflow…
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…
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.
