Classical Bin Packing Research Articles

On-line bin-stretching

We are given a sequence of items that can be packed into m unit size bins. In the classical bin packing problem we fix the size of the bins and try to pack the items in the minimum number of such bins. In contrast, in the bin-stretching problem we fix the number of bins and try to pack the items while stretching the size of the bins as least as possible. We present two on-line algorithms for the bin-stretching problem that guarantee a stretching factor of 5/3 for any number m of bins. We then combine the two algorithms and design an algorithm whose stretching factor is 1.625 for any m. The analysis for the performance of this algorithm is tight. The best lower bound for any algorithm is 4/3 for any m⩾2. We note that the bin-stretching problem is also equivalent to the classical scheduling (load balancing) problem in which the value of the makespan (maximum load) is known in advance.

Polynomial time approximation schemes for class-constrained packing problems

We consider variants of the classic bin packing and multiple knapsack problems, in which sets of items of different classes (colours) need to be placed in bins; the items may have different sizes and values. Each bin has a limited capacity, and a bound on the number of distinct classes of items it can accommodate. In the class-constrained multiple knapsack (CCMK) problem, our goal is to maximize the total value of packed items, whereas in the class-constrained bin-packing (CCBP), we seek to minimize the number of (identical) bins, needed for packing all the items. We give a polynomial time approximation scheme (PTAS) for CCMK and a dual PTAS for CCBP. We also show that the 0–1 class-constrained knapsack admits a fully polynomial time approximation scheme, even when the number of distinct colours of items depends on the input size. Finally, we introduce the generalized class-constrained packing problem (GCCP), where each item may have more than one colour. We show that GCCP is APX-hard, already for the case of a single knapsack, where all items have the same size and the same value. Our optimization problems have several important applications, including storage management for multimedia systems, production planning, and multiprocessor scheduling. Copyright © 2001 John Wiley & Sons, Ltd.

An Approximation Scheme for Bin Packing with Conflicts

In this paper we consider the following bin packing problem with conflicts. Given a set of items V = {1,..., n} with sizes s1,..., s ∈ (0,1) and a conflict graph G = (V, E), we consider the problem to find a packing for the items into bins of size one such that adjacent items (j, j′) ∈ E are assigned to different bins. The goal is to find an assignment with a minimum number of bins. This problem is a natural generalization of the classical bin packing problem. We propose an asymptotic approximation scheme for the bin packing problem with conflicts restricted to d-inductive graphs with constant d. This graph class contains trees, grid graphs, planar graphs and graphs with constant treewidth. The algorithm finds an assignment for the items such that the generated number of bins is within a factor of (1 + ∈) of optimal provided that the optimum number is sufficiently large. The running time of the algorithm is polynomial both in n and in $$\tfrac{1}{\varepsilon}$$ .

The FFD algorithm for the bin packing problem with kernel items

The FFD algorithm is one of the most famous algorithms for the classical bin packing problem. In this paper, some versions of the FFD algorithm are considered in several bin packing problems. Especially, two of them applied to the bin packing problem with kernel items are analyzed. Tight worst-case performance ratios are obtained.

There is no asymptotic PTAS for two-dimensional vector packing

The d-dimensional vector packing problem is a well-known generalization of the classical bin packing problem: For a given list of vectors in [0, 1] d , the goal is to partition the list into the minimum number of subsets, such that in every subset the sum of all vectors is at most one in every coordinate. For the case d = 1, Fernandez de la Vega and Lueker (1981) designed an asymptotic polynomial time approximation scheme. In this note we prove that already for the case d = 2, the existence of an asymptotic polynomial time approximation scheme would imply P = NP. The proof is very simple and uses no new ideas.

On the Asymptotic Worst Case Behavior of Harmonic Fit

In the parametric bin packing problem we must pack a list of items with size smaller than or equal to 1/rin a minimal number of unit-capacity bins. Among the approximation algorithms, the class of Harmonic Fit algorithms (HFM) plays an important role. Lee and Lee (J. Assoc. Comput. Mach.32(1985), 562–572) and Galambos (Ann. Univ. Sci. Budapest Sect. Comput.9(1988), 121–126) provide upper bounds for the asymptotic worst case ratio ofHFMand show tightness for certain values of the parameterM. In this paper we provide worst case examples that meet the known upper bound for additional values ofM, and we show that for remaining values ofMthe known upper bound is not tight. For the classical bin packing problem (r=1), we prove an asymptotic worst case ratio of 12/7 for the caseM= and 1.7 for the caseM=5. We give improved lower bounds for some interesting cases that are left open.

On-line bin packing ? A restricted survey

In the classical bin packing problem, one is asked to pack items of various sizes into the minimum number of equal-sized bins. In the on-line version of this problem, the packer is given the items one by one and must immediately and irrevocably assign every item to its bin, without knowing the future items. Beginning with the first results in the early 1970's, we survey — from the worst case point of view — the approximation results obtained for on-line bin packing, higher dimensional versions of the problem, lower bounds on worst case ratios and related results.

Worst-Case Analysis of Heuristics for the Bin Packing Problem with General Cost Structures

We consider the famous bin packing problem where a set of items must be stored in bins of equal capacity. In the classical version, the objective is to minimize the number of bins used. Motivated by several optimization problems that occur in the context of the storage of items, we study a more general cost structure where the cost of a bin is a concave function of the number of items in the bin. The objective is to store the items in such a way that total cost is minimized. Such cost functions can greatly alter the way the items should be assigned to the bins. We show that some of the best heuristics developed for the classical bin packing problem can perform poorly under the general cost structure. On the other hand, the so-called next-fit increasing heuristic has an absolute worst-case bound of no more than 1.75 and an asymptotic worst-case bound of 1.691 for any concave and monotone cost function. Our analysis also provides a new worst-case bound for the well studied next-tit decreasing heuristic when the objective is to minimize the number of bins used.

Probabilistic analysis of algorithms for dual bin packing problems

In the dual bin packing problem, the objective is to assign items of given size to the largest possible number of bins, subject to the constraint that the total size of the items assigned to any bin is at least equal to 1. We carry out a probabilistic analysis of this problem under the assumption that the items are drawn independently from the uniform distribution on [0, 1] and reveal the connection between this problem and the classical bin packing problem as well as to renewal theory.

Analysis of a Compound Bin Packing Algorithm

Consider the classic bin packing problem, in which we seek to pack a list of items into the minimum number of unit-capacity bins. The worst-case performance of a compound bin packing algorithm that selects the better packing produced by two previously analyzed heuristics, namely, FFD (first fit decreasing) and B2F (best two fit) is investigated. FFD and B2F can asymptotically require as many as $\frac{11}{9}$ and $\frac{5}{4}$ times the optimal number of bins, respectively. A new technique, weighting function averaging, is introduced to prove that our compound algorithm is superior to the individual heuristics on which it is based, never using more than $\frac{6}{5}$ times the optimal number of bins.

Models Arising from a Flexible Manufacturing Machine, Part II: Minimization of the Number of Switching Instants

This paper deals with a job scheduling problem for a flexible manufacturing machine. We consider the same manufacturing environment described in a companion study (Part I), but we use a different performance criterion: to minimize the total number of instants at which tools are switched. This performance criterion is appropriate when the switching time is roughly constant and independent of the number of tool switches. We show that this scheduling problem generalizes the classical bin packing problem, and we present a non-LP-based branch-and-bound procedure that terminates with an optimal solution. We extend this procedure to the case of K machines in sequence.

Anomalous behavior in bin packing algorithms

In the classical bin packing problem one is given a list of items and asked to pack them into the fewest possible unit-sized bins. Given two lists, L 1 and L 2, where L 2 is derived from L 1 by deleting some elements of L 1 and/or reducing the size of some elements of L 1, one might hope that an approximation algorithm would use no more bins to pack L 2 than it uses to pack L 1. Johnson and Graham have given examples showing that First-Fit and First-Fit Decreasing can actually use more bins to pack L 2 than L 1. Graham has also studied this type of behavior among multiprocessor scheduling algorithms. In the present paper we extend this study of anomalous behavior to a broad class of approximation algorithms for bin packing. To do this we introduce a technique which allows one to characterize the monotonic/anomalous behavior of any algorithm in a large, natural class. We then derive upper and lower bounds on the anomalous behavior of the algorithms which are anomalous and provide conditions under which a normally nonmonotonic algorithm becomes monotonic.

Online variable-sized bin packing

The classical bin packing problem is one of the best-known and most widely studied problems of combinatorial optimization. Efficient offline approximation algorithms have recently been designed and analyzed for the more general and realistic model in which bins of differing capacities are allowed (Friesen and Langston (1986)). In this paper, we consider fast online algorithms for this challenging model. Selecting either the smallest or the largest available bin size to begin a new bin as pieces arrive turns out to yield a tight worst-case ratio of 2. We devise a slightly more complicated scheme that uses the largest available bin size for small pieces, and selects bin sizes for large pieces based on a user-specified fill factor ƒ≥ 1 2 , and prove that this strategy guarantees a worst-case bound not exceeding 1.5+ ƒ 2 .

An Efficient Approximation Scheme for Variable-Sized Bin Packing

In the classical bin packing problem one is required to pack a given list of items into the smallest possible number of unit-sized bins. Because this problem is NP-complete, researchers have tried to find efficient approximation algorithms that solve this problem in a reasonable amount of time. If we let $A(I)$ be the number of bins used by algorithm A to pack a list of items I, and ${\textit{OPT}}(I)$ be the minimum number of bins necessary to pack I, one defines the asymptotic performance ratio of A to be ${{A(I)} / {{\textit{OPT}}(I)}}$, as ${\textit{OPT}}(I)$ tends to infinity. De la Vega and Lueker presented an approximation scheme, which for any $ \in > 0$, yields an approximation algorithm with performance ratio $1 + \varepsilon $. This scheme has time complexity polynomial in n, the number of items, but exponential in ${1 / \varepsilon }$. Karmarkar and Karp extended this scheme to one that has time complexity polynomial in n and ${1 / \varepsilon }$. In the variable-sized bin packing problem, one...

Variable Sized Bin Packing

In the classical bin packing problem one seeks to pack a list of pieces in the minimum space using unit capacity bins. This paper addresses the more general problem in which a fixed collection of bin sizes is allowed. Three efficient approximation algorithms are described and analyzed. They guarantee asymptotic worst-case performance bounds of 2, ${3 / 2}$ and ${4 / 3}$.