Bin packing problem
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. Computationally, the problem is NP-hard, and the corresponding decision problem - deciding if items can fit into a specified number of bins - is NP-complete. Despite its  worst-case hardness, optimal solutions to very large instances of the problem can be produced with sophisticated algorithms. In addition, many approximation algorithms exist. For example, the first fit algorithm provides a fast but often non-optimal solution, involving placing each item into the first bin in which it will fit. It requires "Θ"("n" log "n") time, where "n" is the number of items to be packed. The algorithm can be made much more effective by first sorting the list of items into decreasing order (sometimes known as the first-fit decreasing algorithm), although this still does not guarantee an  optimal solution, and for longer lists may increase the running time of the algorithm. It is known, however, that there always exists at least one ordering of items that allows first-fit to produce an optimal solution. There are many variations of this problem, such as 2D packing, linear packing, packing by weight, packing by cost, and so on. The bin packing problem can also be seen as a special case of the cutting stock problem. When the number of bins is restricted to 1 and each item is characterised by both a volume and a value, the problem of  maximizing the value of items that can fit in the bin is known as the knapsack problem. A variant of bin packing that occurs in practice is when items can share space when packed into a bin. Specifically, a set of items could occupy less space when packed together than the sum of their individual sizes. This variant is known as VM packing since when virtual machines (VMs) are packed in a server, their total memory requirement could decrease due to pages shared by the VMs that need only be stored once. If items can share space in arbitrary ways , the bin packing problem is hard to even approximate. However, if the space sharing fits into a hierarchy, as is the case with memory sharing in virtual machines, the bin packing problem can be efficiently approximated. Another variant of bin packing of interest in practice is the so-called online bin packing. Here the items of different volume are supposed to arrive sequentially, and the decision maker has to decide whether to select and pack the currently observed item, or else to let it pass. Each decision is without recall. In contrast, offline bin packing allows rearranging the items in  the hope of achieving a better packing once additional items arrive. This of course requires additional storage for holding the items to be rearranged. Formal statement. In "Computers and Intractability" Garey and Johnson list the bin packing problem under the reference [SR1]. They define its decision variant as follows. Instance: Finite set formula_1 of items, a size formula_2 for each formula_3, a positive integer bin capacity formula_4, and a positive integer formula_5. Question: Is there a partition of formula_1 into disjoint sets formula_7 such that the sum of the sizes of the items in each formula_8 is formula_4 or less ? Note that in the literature often an equivalent notation is used, where formula_10 and formula_11 for each formula_3. Furthermore, research is mostly interested in the optimization variant, which asks for the smallest possible value of formula_5. A solution is "optimal" if it has minimal formula_5. The formula_5-value for an optimal solution for a set of items formula_1 is denoted by formula_17 or just formula_18 if the set of items is clear from the context. A possible integer linear programming formulation of the problem is: where formula_19 if bin formula_20 is used and formula_21 if item formula_22 is put into  bin formula_20. Hardness of bin packing. The bin packing problem is strongly NP-complete. This can be proven by reducing the strongly NP-complete 3-partition problem to bin packing. Furthermore, there can be no approximation algorithm with absolute approximation ratio smaller than formula_24 unless formula_25. This can be proven by a reduction from the partition problem: given an instance of Partition where the sum of all input numbers is 2 "T", construct an instance of bin-packing in which the bin size is "T". If there exists an equal partition of the inputs, then the optimal packing needs 2  bins; therefore, every algorithm with approximation ratio smaller than 3/2 must return less than 3 bins, which must be 2 bins. In contrast, if there is no equal partition of the inputs, then the optimal packing needs at least 3 bins. On the other hand, bin packing is solvable in pseudo-polynomial time for any fixed number of bins formula_26, and solvable in polynomial time for any fixed bin capacity formula_4. Approximation algorithms for bin packing. To measure the performance of an approximation algorithm there are two approximation ratios considered in the literature. For a given list of items formula_28  the number formula_29 denotes the number of bins used when algorithm formula_30 is applied to list formula_28, while formula_32 denotes the optimum number for this list. The absolute worst-case performance ratio formula_33 for an algorithm formula_30 is defined as On the other hand, the asymptotic worst-case ratio formula_36 is defined as Equivalently, formula_36 is the smallest number such that, for some constant "K," for all lists "L:" Additionally, one can restrict the lists to those for which all items have a size of at most formula_40. For such lists, the bounded size performance ratios are denoted as formula_41  and formula_42. Approximation algorithms for bin packing can be classified into two categories: Online heuristics. In the online version of the bin packing problem, the items arrive one after another and the (irreversible) decision where to place an item has to be made before knowing the next item or even if there will be another one. A diverse set of offline and online heuristics for bin-packing have been studied by David S. Johnson on his Ph.D. thesis. Single-class algorithms. There are many simple algorithms that use the following general scheme: The algorithms differ in the criterion by which  they choose the open bin for the new item in step 1 (see the linked pages for more information): In order to generalize these results, Johnson introduced two classes of online heuristics called "any-fit algorithm" and "almost-any-fit" algorithm: formula_76. formula_78 In particular: formula_79. Refined algorithms. Better approximation ratios are possible with heuristics that are not AnyFit. These heuristics usually keep several classes of open bins, devoted to items of different size-ranges (see the linked pages for more information): General lower bounds for online algorithms. Yao proved 1980 that there can be no online algorithm with asymptotic  competitive ratio smaller than formula_24. Brown and Liang improved this bound to formula_105. Afterward, this bound was improved to formula_106 by Vliet. In 2012, this lower bound was again improved by Békési and Galambos to formula_107. Offline algorithms. In the offline version of bin packing, the algorithm can see all the items before starting to place them into bins. This allows to attain improved approximation ratios. Multiplicative approximation. The simplest technique used by offline algorithms is: Johnson proved that any AnyFit algorithm A that runs on a list ordered by descending size has an asymptotic approximation ratio offormula_108.Some algorithms in  this family are (see the linked pages for more information): Fernandez de la Vega and Lueker presented a PTAS for bin packing. For every formula_111, their algorithm finds a solution with size at most formula_112 and runs in time formula_113, where formula_114 denotes a function only dependent on formula_44. For this algorithm, they invented the method of "adaptive input rounding": the input numbers are grouped and rounded up to the value of the maximum in each group. This yields an instance with a small number of different sizes, which can be solved exactly using the configuration linear program. Additive approximation . The Karmarkar-Karp bin packing algorithm finds a solution with size at most formula_116, and runs in time polynomial in formula_117 (the polynomial has a high degree - at least 8). Rothvoss presented an algorithm that generates a solution with at most formula_118 bins. Hoberg and Rothvoss improved this algorithm to generate a solution with at most formula_119 bins. The algorithm is randomized, and its running-time is polynomial in formula_117. Exact algorithms. Martello and Toth developed an exact algorithm for the 1-dimensional bin-packing problem, called MTP. A faster alternative is the Bin Completion algorithm proposed by Korf in  2002 and later improved. A further improvement was presented by Schreiber and Korf in 2013. The new Improved Bin Completion algorithm is shown to be up to five orders of magnitude faster than Bin Completion on non-trivial problems with 100 items, and outperforms the BCP (branch-and-cut-and-price) algorithm by Belov and Scheithauer on problems that have fewer than 20 bins as the optimal solution. Which algorithm performs best depends on problem properties like number of items, optimal number of bins, unused space in the optimal solution and value precision. Small number of different sizes. A special  case of bin packing is when there is a small number "d" of different item sizes. There can be many different items of each size. This case is also called "high-multiplicity bin packing", and It admits more efficient algorithms than the general problem. Cardinality constraints on the bins. There is a variant of bin packing in which there are cardinality constraints on the bins: each bin can contain at most "k" items, for some fixed integer "k". Non-additive functions. There are various ways to extend the bin-packing model to more general cost and load functions: Related problems . In the bin packing problem, the "size" of the bins is fixed and their "number" can be enlarged (but should be as small as possible). In contrast, in the multiway number partitioning problem, the "number" of bins is fixed and their "size" can be enlarged. The objective is to find a partition in which the bin sizes are as nearly equal is possible (in the variant called multiprocessor scheduling problem or minimum makespan problem, the goal is specifically to minimize the size of the largest bin). In the inverse bin packing problem, both the number of bins and their sizes  are fixed, but the item sizes can be changed. The objective is to achieve the minimum perturbation to the item size vector so that all the items can be packed into the prescribed number of bins. In the maximum resource bin packing problem, the goal is to "maximize" the number of bins used, such that, for some ordering of the bins, no item in a later bin fits in an earlier bin. In a dual problem, the number of bins is fixed, and the goal is to minimize the total number or the total size of items placed into the  bins, such that no remaining item fits into an unfilled bin. In the bin covering problem, the bin size is bounded "from below": the goal is to "maximize" the number of bins used such that the total size in each bin is at least a given threshold. In the fair indivisible chore allocation problem (a variant of fair item allocation), the items represents chores, and there are different people each of whom attributes a different difficulty-value to each chore. The goal is to allocate to each person a set of chores with an upper bound on its total difficulty -value (thus, each person corresponds to a bin). Many techniques from bin packing are used in this problem too. In the guillotine cutting problem, both the items and the "bins" are two-dimensional rectangles rather than one-dimensional numbers, and the items have to be cut from the bin using end-to-end cuts. In the selfish bin packing problem, each item is a player who wants to minimize its cost. There is also a variant of bin packing in which the cost that should be minimized is not the number of bins, but rather a certain concave function of  the number of items in each bin. Other variants are two-dimensional bin packing, three-dimensional bin packing, bin packing with delivery,