Problems which attempt to find the most efficient way to pack objects into containers
This article is about geometric packing problems. For numerical packing problems, see Knapsack problem.
Spheres or circles packed loosely (top) and more densely (bottom)
Covering/packing-problem pairs
Covering problems
Packing problems
Minimum set cover
Maximum set packing
Minimum edge cover
Maximum matching
Minimum vertex cover
Maximum independent set
Bin covering
Bin packing
Polygon covering
Rectangle packing
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Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap.
In a bin packing problem, people are given:
A container, usually a two- or three-dimensional convex region, possibly of infinite size. Multiple containers may be given depending on the problem.
A set of objects, some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or a single object of a fixed dimension that can be used repeatedly.
Usually the packing must be without overlaps between goods and other goods or the container walls. In some variants, the aim is to find the configuration that packs a single container with the maximal packing density. More commonly, the aim is to pack all the objects into as few containers as possible.[1] In some variants the overlapping (of objects with each other and/or with the boundary of the container) is allowed but should be minimized.
^Lodi, A.; Martello, S.; Monaci, M. (2002). "Two-dimensional packing problems: A survey". European Journal of Operational Research. 141 (2). Elsevier: 241–252. doi:10.1016/s0377-2217(02)00123-6.
Packingproblems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to...
The bin packingproblem is an optimization problem, in which items of different sizes must be packed into a finite number of bins or containers, each of...
sphere packingproblems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions...
Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems. Suppose...
knapsack problem Cutting stock problem – Mathematical problem in operations research Knapsack auction List of knapsack problemsPackingproblem – Problems which...
Rectangle packing is a packingproblem where the objective is to determine whether a given set of small rectangles can be placed inside a given large polygon...
Square packing is a packingproblem where the objective is to determine how many congruent squares can be packed into some larger shape, often a square...
space to the volume of the space itself. In packingproblems, the objective is usually to obtain a packing of the greatest possible density. If K1,......
Discrete Ham-Sandwich Theorems: Provably Good Algorithms for Routing and PackingProblems". UC Berkeley. Retrieved 19 May 2014. Advisor: Clark D. Thompson "Prabhakar...
a sealing material Packingproblems, a family of optimization problems in mathematics All pages with titles beginning with Packing All pages with titles...
Circle packing in a square is a packingproblem in recreational mathematics, where the aim is to pack n unit circles into the smallest possible square...
sphere packings thanks to their large number. Sphere packingproblems are distinguished between packings in given containers and free packings. This article...
The strip packingproblem is a 2-dimensional geometric minimization problem. Given a set of axis-aligned rectangles and a strip of bounded width and infinite...
classification of problem-solving tasks is into well-defined problems with specific obstacles and goals, and ill-defined problems in which the current...
and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of...
Circle packing in a circle is a two-dimensional packingproblem with the objective of packing unit circles into the smallest possible larger circle. If...
hexagonal packing is generally not optimal for small numbers of circles. Specific problems of this type that have been studied include: Circle packing in a...
that. Covering problems are minimization problems and usually integer linear programs, whose dual problems are called packingproblems. The most prominent...
cover problem. Benchmarks with Hidden Optimum Solutions for Set Covering, Set Packing and Winner Determination A compendium of NP optimization problems -...
Sphere packing in a sphere is a three-dimensional packingproblem with the objective of packing a given number of equal spheres inside a unit sphere. It...
space such that they each touch a common unit sphere. For a given sphere packing (arrangement of spheres) in a given space, a kissing number can also be...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer...
In geometry, tetrahedron packing is the problem of arranging identical regular tetrahedra throughout three-dimensional space so as to fill the maximum...
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