Next-fit-decreasing (NFD) is an algorithm for bin packing. Its input is a list of items of different sizes. Its output is a packing - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem. The NFD algorithm uses the following heuristic:
In short: NFD orders the items by descending size, and then calls next-fit bin packing.
Baker and Coffman [1] proved that, for every integer r, when the size of all items is at most 1/r, the asymptotic approximation ratio of RFD satisfies
,
where is a sequence whose first elements are approximately 1.69103, 1.42312, 1.30238. In particular, taking r=1 implies that
.
Later, NFD has also been analyzed probabilistically. [2]
Next-Fit packs a list and its inverse into the same number of bins. Therefore, Next-Fit-Increasing has the same performance as Next-Fit-Decreasing. [3]
However, Next-Fit-Increasing performs better when there are general cost structures. [4]
The bin packing problem is an optimization problem, in which items of different sizes must be packed into a finite number of bins or containers, each of a fixed given capacity, in a way that minimizes the number of bins used. The problem has many applications, such as filling up containers, loading trucks with weight capacity constraints, creating file backups in media and technology mapping in FPGA semiconductor chip design.
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that
Guillotine cutting is the process of producing small rectangular items of fixed dimensions from a given large rectangular sheet, using only guillotine-cuts. A guillotine-cut is a straight bisecting line going from one edge of an existing rectangle to the opposite edge, similarly to a paper guillotine.
Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive update rules of stochastic approximation methods can be used, among other things, for solving linear systems when the collected data is corrupted by noise, or for approximating extreme values of functions which cannot be computed directly, but only estimated via noisy observations.
Job-shop scheduling, the job-shop problem (JSP) or job-shop scheduling problem (JSSP) is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. In a general job scheduling problem, we are given n jobs J1, J2, ..., Jn of varying processing times, which need to be scheduled on m machines with varying processing power, while trying to minimize the makespan – the total length of the schedule. In the specific variant known as job-shop scheduling, each job consists of a set of operationsO1, O2, ..., On which need to be processed in a specific order. Each operation has a specific machine that it needs to be processed on and only one operation in a job can be processed at a given time. A common relaxation is the flexible job shop, where each operation can be processed on any machine of a given set.
The vertex k-center problem is a classical NP-hard problem in computer science. It has application in facility location and clustering. Basically, the vertex k-center problem models the following real problem: given a city with facilities, find the best facilities where to build fire stations. Since firemen must attend any emergency as quickly as possible, the distance from the farthest facility to its nearest fire station has to be as small as possible. In other words, the position of the fire stations must be such that every possible fire is attended as quickly as possible.
The strip packing problem is a 2-dimensional geometric minimization problem. Given a set of axis-aligned rectangles and a strip of bounded width and infinite height, determine an overlapping-free packing of the rectangles into the strip minimizing its height. This problem is a cutting and packing problem and is classified as an Open Dimension Problem according to Wäscher et al.
In computer science, multiway number partitioning is the problem of partitioning a multiset of numbers into a fixed number of subsets, such that the sums of the subsets are as similar as possible. It was first presented by Ronald Graham in 1969 in the context of the Identical-machines scheduling problem. The problem is parametrized by a positive integer k, and called k-way number partitioning. The input to the problem is a multiset S of numbers, whose sum is k*T.
In the bin covering problem, items of different sizes must be packed into a finite number of bins or containers, each of which must contain at least a certain given total size, in a way that maximizes the number of bins used.
The multifit algorithm is an algorithm for multiway number partitioning, originally developed for the problem of identical-machines scheduling. It was developed by Coffman, Garey and Johnson. Its novelty comes from the fact that it uses an algorithm for another famous problem - the bin packing problem - as a subroutine.
Identical-machines scheduling is an optimization problem in computer science and operations research. We are given n jobs J1, J2, ..., Jn of varying processing times, which need to be scheduled on m identical machines, such that a certain objective function is optimized, for example, the makespan is minimized.
Next-fit is an online algorithm for bin packing. Its input is a list of items of different sizes. Its output is a packing - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem. The next-fit algorithm uses the following heuristic:
Best-fit is an online algorithm for bin packing. Its input is a list of items of different sizes. Its output is a packing - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem. The best-fit algorithm uses the following heuristic:
First-fit (FF) is an online algorithm for bin packing. Its input is a list of items of different sizes. Its output is a packing - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem. The first-fit algorithm uses the following heuristic:
Harmonic bin-packing is a family of online algorithms for bin packing. The input to such an algorithm is a list of items of different sizes. The output is a packing - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem.
First-fit-decreasing (FFD) is an algorithm for bin packing. Its input is a list of items of different sizes. Its output is a packing - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity. Ideally, we would like to use as few bins as possible, but minimizing the number of bins is an NP-hard problem, so we use an approximately-optimal heuristic.
Longest-processing-time-first (LPT) is a greedy algorithm for job scheduling. The input to the algorithm is a set of jobs, each of which has a specific processing-time. There is also a number m specifying the number of machines that can process the jobs. The LPT algorithm works as follows:
Balanced number partitioning is a variant of multiway number partitioning in which there are constraints on the number of items allocated to each set. The input to the problem is a set of n items of different sizes, and two integers m, k. The output is a partition of the items into m subsets, such that the number of items in each subset is at most k. Subject to this, it is required that the sums of sizes in the m subsets are as similar as possible.
The configuration linear program (configuration-LP) is a particular linear programming used for solving combinatorial optimization problems. It was introduced in the context of the cutting stock problem. Later, it has been applied to bin packing and job scheduling. In the configuration-LP, there is a variable for each possible configuration - each possible multiset of items that can fit in a single bin. Usually, the number of configurations is exponential in the problem size, but in some cases it is possible to attain approximate solutions using only a polynomial number of configurations.
High-multiplicity bin packing is a special case of the bin packing problem, in which the number of different item-sizes is small, while the number of items with each size is large. While the general bin-packing problem is NP-hard, the high-multiplicity setting can be solved in polynomial time, assuming that the number of different sizes is a fixed constant.
