Partitioning bin-packing algorithms for distributed real-time systems

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.

Multi-capacity bin-packing is a generalization of the classical one-dimensional bin-packing problem in which the bin capacity and the item sizes are represented by d-dimensional vectors. Previous work in d-capacity bin-packing algorithms analyzed variants of single capacity bin-packing, extended to deal with the d capacities. These extended algorithms are oblivious to the additional capacity information, however, and do not scale well with increasing d. We provide new packing algorithms which use the additional capacity information to provide better packing and show how these algorithms might lead to better multi-resource allocation and scheduling solutions.

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

We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axis-parallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the approximation ratio of 2 which was established more than a decade ago. Moreover, we present a polynomial time approximation scheme ($\mathcal{PTAS}$) for strip packing where rotations by 90 degrees are permitted and an algorithm for two-dimensional bin packing with an absolute worst-case ratio of 2, which is optimal provided $\mathcal{P} \not= \mathcal{NP}$.

A two-dimensional bin packing model was established to solve the problem of goods placement in the material box, which was optimized to improve space utilization. By discretization of the model, a bin packing algorithm with left-bottom priority is designed. By adopting the improved genetic algorithm with the optimal individual retention strategy, the optimal sequence of various kinds of goods in the material box is obtained. Two types of experiments of multi-algorithm comparison simulation and multi-solution simulation are designed. The first type of experimental results show that the algorithm has some advantages in fitness and convergence speed, and the second type of experimental results show that it can provide multiple optimal solutions for solving small-scale bin packing problems.

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

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.

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

The study on one-dimensional bin packing problem may bring about many important applications such as multiprocessor scheduling, resource allocating, real-world planning and packing. Harmonic algorithms (including H K, RH, etc.) for bin packing have been famous for their linear-time and on-line properties for a long time. This paper profoundly analyzes the average-case performance of harmonic algorithms for pieces of i.i.d. sizes, provides the average-case performance ratio of H K under (0,d] (d ≤ 1) uniform distribution and the average-case performance ratio of RH under (0,1] uniform distribution. We also finished the discussion of the worst-case performance ratio of RH. Moreover, we propose a new improved version of RH that obtains better worst- and average-case performance ratios.

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.

This paper integrates two-dimensional bin packing with facility layout concepts to address scenarios in which items must not only fit within a confined space but also be arranged according to spatial priorities. We embed a prioritization matrix into the bin packing framework, enabling items to be clustered with one another or pulled toward certain bin access points based on assigned priority weights. Unlike traditional bin packing, which focuses on space utilization alone, our approach balances proximity to bin access points and adjacency among functionally related items, extending the utility of bin packing to applications requiring more nuanced layout preferences. We introduce a single mixed-integer linear programming (MILP) model and a complementary sliding window decomposition method that scales effectively to larger problem instances. Numerical experiments illustrate that this decomposition approach consistently outperforms a direct MILP solve with a commercial solver in both runtime and solution quality. This computational study underscores the flexibility and effectiveness of embedding multi-level priorities into bin packing.

Cloud computing systems are the backbone of our technology needs in everyday life and are one of the major electric energy consumers globally. Any improvement that can be added to the energy efficiency of these vast systems constitutes a big gain worldwide in our never-ending battle with climate change and pollution. This paper proposes a new algorithm for task scheduling in Cloud systems based on the bin packing algorithm using a greedy implementation in conjunction with an optimization algorithm for resource task execution. By executing tasks in as much time as possible without impacting customer experience too much due to latency and by using resources at close to 100% in order to use the minimum number of servers we can achieve on average a 3.79% increase in energy efficiency. On top of that, our algorithm is robust to extreme variations of incoming task deadline distributions.

In this paper, the NP-complete problem of 1D bin packing is considered. In the 'bin packing problem', a set of numbers or elements that need to be loaded in the bins is considered. The total load of the elements in any bin must not exceed the bin size or bin capacity. These elements must be packed or loaded in the bins in such a way that there is minimum amount of space left in each bin and the number of bins to be used is minimised. A new algorithm, which helps in tackling the bin packing problem, by giving optimal solutions in minimum time, is presented. The algorithms currently in use, requires the elements to be sorted before the loading. The algorithm discussed in this paper avoids the sorting operation before the loading while giving the same best fit results.

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.