Probabilistic Analysis of Bin Packing Heuristics

Approximation and online algorithms for multidimensional bin packing: A survey

Summary. The bin packing problem is a fundamental and important optimization problem in theoretical computer science [4], [6]. An instance is a sequence of items, each being of positive size at most one. The task is to place all the items into bins so that the total size of items in each bin is at most one and the number of bins that contain at least one item is minimum. Approximation algorithms have been intensively studied. Algorithm NextFit would be the simplest one. The algorithm repeatedly does the following: If the first unprocessed item in the sequence can be placed, in terms of size, additionally to the bin into which the algorithm has placed an item the last time, place the item into that bin; otherwise place the item into an empty bin. Johnson [5] proved that the number of the resulting bins by algorithm NextFit is less than twice the number of the fewest bins that are needed to contain all items. In this article, we formalize in Mizar [1], [2] the bin packing problem as follows: An instance is a sequence of positive real numbers that are each at most one. The task is to find a function that maps the indices of the sequence to positive integers such that the sum of the subsequence for each of the inverse images is at most one and the size of the image is minimum. We then formalize algorithm NextFit, its feasibility, its approximation guarantee, and the tightness of the approximation guarantee.

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 $$\tfrac{1}{\varepsilon}$$ .

A study on load-balanced variants of the bin packing problem

Best Fit is a well known online algorithm for the bin packing problem, where a collection of one-dimensional items has to be packed into a minimum number of unit-sized bins. In a seminal work, Kenyon [SODA 1996] introduced the (asymptotic) random order ratio as an alternative performance measure for online algorithms. Here, an adversary specifies the items, but the order of arrival is drawn uniformly at random. Kenyon’s result establishes lower and upper bounds of 1.08 and 1.5, respectively, for the random order ratio of Best Fit. Although this type of analysis model became increasingly popular in the field of online algorithms, no progress has been made for the Best Fit algorithm after the result of Kenyon. We study the random order ratio of Best Fit and tighten the long-standing gap by establishing an improved lower bound of 1.10. For the case where all items are larger than 1/3, we show that the random order ratio converges quickly to 1.25. It is the existence of such large items that crucially determines the performance of Best Fit in the general case. Moreover, this case is closely related to the classical maximum-cardinality matching problem in the fully online model. As a side product, we show that Best Fit satisfies a monotonicity property on such instances, unlike in the general case. In addition, we initiate the study of the absolute random order ratio for this problem. In contrast to asymptotic ratios, absolute ratios must hold even for instances that can be packed into a small number of bins. We show that the absolute random order ratio of Best Fit is at least 1.3. For the case where all items are larger than 1/3, we derive upper and lower bounds of 21/16 and 1.2, respectively.

We consider the bin packing problem in the non-cooperative game setting. In the game there are a set of items with sizes between 0 and 1 and a number of bins each with a capacity of 1. Each item seeks to be packed in one of the bins so as to minimize its cost (payoff). The social cost is the number of bins used in the packing. Existing research has focused on three bin packing games with selfish items, namely the Unit game, the Proportional game, and the General Weight game, each of which uses a unique payoff rule. In this paper we propose a new bin packing game in which the payoff of an item is a function of its own size and the size of the maximum item in the same bin. We find that the new payoff rule induces the items to reach a better Nash equilibrium. We show that the price of anarchy of the new bin packing game is \(\frac{3}{2}\) and prove that any feasible packing can converge to a Nash equilibrium in \(n^2-n\) steps without increasing the social cost.

Construction heuristics for two-dimensional irregular shape bin packing with guillotine constraints

In this paper, we study extensions to bin packing algorithms to pack software modules into processors in real-time systems. We refer to this approach as Partitioning Bin-Packing. In this study, we analytically show that with partitioning bin-packing techniques the number of bins required by traditional bin packing can be reduced. We also evaluate heuristics to minimise both the number of processors (bins) needed and the network bandwidth required by communicating software modules that are partitioned across different processors. We find that a significant reduction in the number of bins is possible. Finally, different heuristics lead to different tradeoffs in processing vs. network needs.

Probabilistic analysis of algorithms for dual bin packing problems

The more-dimensional bin packing problem (BPP) considered here requires packing a set of rectangular-shaped items into a minimum number of identical rectangular-shaped bins. All items may be rotated and the guillotine cut constraint has to be respected. A straightforward heuristic is presented that is based on a method for the container loading problem following a wall-building approach and on a method for the one-dimensional BPP. 1,800 new benchmark instances are introduced for the two-dimensional and three-dimensional BPP. The instances include more than 1,500 items on average. Applied to these very large instances, the heuristic generates solutions of acceptable quality in short computation times. Moreover, the influence of different instance parameters on the solution quality is investigated by an extended computational study.

We are given a set of items, each characterized by a weight and a fragility, and a large number of uncapacitated bins. Our aim is to find the minimum number of bins needed to pack all items, in such a way that in each bin the sum of the item weights is less than or equal to the smallest fragility of an item in the bin. The problem is known in the literature as the Bin Packing Problem with Fragile Objects, and appears in the telecommunication field, when one has to assign cellular calls to available channels by ensuring that the total noise in a channel does not exceed the noise acceptance limit of a call.We propose several techniques to compute lower and upper bounds for this problem. For what concerns lower bounds, we present combinatorial techniques with guaranteed worst case and a more complex bound based on a column generation algorithm. We also present a technique to compute, in a fast heuristic way, dual information that is used to strengthen the convergence of the column generation. For what concerns upper bounds, we present a large set of constructive heuristics followed by a Variable Neighborhood Search algorithm. Our heuristic techniques are aimed at both computing upper bounds and strengthening the behavior of the lower bounds in a matheuristic fashion. Extensive computational tests show the effectiveness of the proposed algorithms.

The bin packing problem is to pack a list of reals in (0, 1] into unit-capacity bins using the minimum number of bins. LetR∞[A] be the limiting worst value for the ratioA(L)/OPT(L) asOPT(L) goes to ∞, whereA(L) denotes the number of bins used when the approximation algorithmA is applied to the listL, andOPT(L) denotes the minimum number of bins needed to packL. For Best-k-Fit (BkF for short,k≥1), we prove in this paper thatR∞[BkF]=1.7+3/10k.

We consider a variant of the classical one-dimensional bin-packing problem: The number of bins is fixed and the object is to maximize the number of pieces packed from some given set. Both problems have applications in processor and storage allocation in computer systems in addition to a broad application in operations research. It can easily be shown that both problems are NP-complete; our approach will be to propose and analyze very fast heuristics. We consider a class of algorithms and bound the performance of an arbitrary algorithm in that class. Finally we propose an algorithm, the first-fit-increasing algorithm, and analyze its running time and relative performance.

The Intermediate Price of Anarchy ([formula omitted]) in bin packing games

For any listL ofn numbers in (0, 1) letL* denote the minimum number of unit capacity bins needed to pack the elements ofL. We prove that, for every positive e, there exists anO(n)-time algorithmS such that, ifS(L) denotes the number of bins used byS forL, thenS(L)/L*≦1+e for anyL providedL* is sufficiently large.