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Topic: Packing problem

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Bin packing problem

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	PENTAGON PACKING IN A CIRCLE
		This problem is analogous with the Tammes problem of the densest packing of equal circles on a sphere.
		— that is analogous to the problem of the densest packing of equal circles in a circle.
		The problems of the densest packing of equal circles in the plane, in a circle and on a sphere are well known.
	members.tripod.com /vismath7/proceedings/tarnai.htm (1329 words)

	Bin Packing
		Suburban residents must be having similar problems because the growth in using off-site storage facilities is on the rise.
		This problem, known as the 1-dimensional bin packing problem, is one of many mathematical packing problems which are of both theoretical and applied interest in mathematics.
		However, in the problem being considered here we are not allowed to have part of a weight in one container and part in another.
	www.ams.org /featurecolumn/archive/bins1.html (305 words)

	Packing problem - Wikipedia, the free encyclopedia
		Packing problems are one area where mathematics meets puzzles (recreational mathematics).
		Usually the packing must be without gaps or overlaps, but in some packing problems the overlapping (of goods with each other and/or with the boundary of the container) is allowed but should be minimised.
		A problem is the square packing problem, where one must determine how many squares of side 1 you can pack into a square of side a.
	en.wikipedia.org /wiki/Packing_problem (601 words)

	Bin Packing and Machine Scheduling
		Packing ads into breaks is an example of a bin packing problem.
		There are many problems which have the flavor of putting weights into bins of fixed size; the goal is to find the smallest number of bins into which the weights will fit.
		Scheduling problems are omnipresent in the operation of all large systems: scheduling classes and finals in schools; scheduling planes, trains, and buses in the transportation sector of the economy; and scheduling machines in the manufacture of the products that we use for everyday life.
	www.ams.org /featurecolumn/archive/packings1.html (234 words)

	Bin Packing
		Packing pieces of a standard jigsaw puzzle is a different problem than packing squares into a rectangular box.
		Packing boxes is much easier than packing arbitrary geometric shapes, enough so that a reasonable approach for general shapes is to pack each part into its own box and then pack the boxes.
		Particularly notorious is the ``Kepler conjecture,'' the apparently still-open problem of establishing the densest packing of unit spheres in three dimensions.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK5/NODE192.HTM (1114 words)

	Bin Packing
		One-dimensional bin packing is a classic problem with many practical applications related to minimization of space or time.
		The goal is to pack a collection of objects into the minimum number of fixed-size "bins".
		Packing is usually improved by starting with the largest objects first.
	www.cs.arizona.edu /icon/oddsends/bpack/bpack.htm (417 words)

	Sphere Packing (Site not responding. Last check: 2007-11-05)
		The so-called sphere packing problem was born in 1611, when the German astronomer Johannes Kepler asked himself which is the most efficient way to pack spheres leaving as few gaps as possible.
		Traditionally mathematicians would simply alter the variables to maximise the packing efficiency for the equation, and then see which arrangement is associated with the variables, however the equation is immensely complicated, which puts the maximisation process beyond paper and pencil calculations, and even challenges the limits of computers.
		This was one of the first significant problems to succumb to the power of computing, and sparked concerns about how such proofs should be checked.
	www.simonsingh.com /Sphere_Packing.html (642 words)

	The Geometry Junkyard: Sphere Packing
		Problems of arranging balls densely arise in many situations, particularly in coding theory (the balls are formed by the sets of inputs that the error-correction would map into a single codeword).
		The Kepler Conjecture on dense packing of spheres.
		Packing circles in the hyperbolic plane, Java animation by Kevin Pilgrim illustrating the effects of changing radii in the hyperbolic plane.
	www.ics.uci.edu /~eppstein/junkyard/spherepack.html (790 words)

	Ecological Bin Packing
		Bin packing, or the placement of objects of certain weights into different bins subject to certain constraints, is an historically interesting problem.
		In this problem you will be solving a bin packing problem that deals with recycling glass.
		The problem is to minimize the number of bottles that are moved.
	acm.uva.es /p/v1/102.html (427 words)

	Set Packing
		Discussion: Set packing problems arise in partitioning applications, where we need to partition elements under strong constraints on what is an allowable partition.
		The key feature of packing problems is that no elements are permitted to be covered by more than one set.
		Set packing is used here to represent a bunch of problems on sets, all of which are NP-complete and all of which are quite similar:
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK5/NODE202.HTM (706 words)

	Problem 55: Pallet Loading
		This problem is not even known to be in NP, because of the compact input description, and the possibly complicated structure of a packing, if there is one.
		Natural packing problem; first-rate example of the relevance of coding input and output.
		Tar92] showed that the problem can be solved in time polynomial in the size of the input if we are restricted to ``guillotine'' patterns, i.e., arrangements of items that can be obtained by a recursive sequence of edge-to-edge cuts.
	maven.smith.edu /~orourke/TOPP/P55.html (213 words)

	Bin packing problem - Wikipedia, the free encyclopedia
		In it, objects of different volumes must be packed into a finite number of bins of capacity V in a way that minimizes the number of bins used.
		There are many variations of this problem, such as 2D packing, linear packing, packing by weight, packing by cost, and so on.
		Although these simple strategies are often good enough, efficient approximation algorithms have been demonstrated that can solve the bin packing problem within any fixed percentage of the optimal solution for sufficiently large inputs (this is called an asymptotic polynomial-time approximation scheme).
	en.wikipedia.org /wiki/Bin_packing_problem (446 words)

	Problem 56: Packing Unit Squares in a Simple Polygon
		The problem is known to be NP-complete for polygons with holes [
		The problem is the decision version for two optimization problems of very different behavior.
		BF01] conjecture the problem to be polynomially solvable.
	maven.smith.edu /~orourke/TOPP/P56.html (164 words)

	Packing with Genetic Algorithms (Site not responding. Last check: 2007-11-05)
		The goal of your program is to provide, for each packing problem, an arrangement of the 10 pieces, on a grid, that minimizes the area of the smallest rectangle that encloses all 10 pieces.
		For each problem, your program should print the input pieces that will be packed.
		Your packing could be generated a la the old computer game Tetris: "drop" the pieces into a "bin".
	www.emunix.emich.edu /~evett/AI/GA-packing.html (539 words)

	Cutting and Packing
		There are several variants of knapsack problems, such as the classical knapsack problem, multiconstrained knapsack problems, multiple knapsack problems, and many others.
		Solving cutting and packing problems can be done in various ways, we developed metaheuristics and exact algorithms for dealing with those problems.
		Weight-codings in a genetic algorithm for the multiconstraint knapsack problem.
	www.ads.tuwien.ac.at /research/CaP (778 words)

	Kepler's Sphere Packing Problem Solved
		The general problem as considered by Kepler and subsequent mathematicians is formulated not in terms of the number of spheres that can be packed together but the density of the packing, i.e., the total volume of the spheres divided by the total volume of the container into which they are packed.
		Moreover, both problems had subtle difficulties that led a number of mathematicians to believe they had found a solution that subsequently turned out to be false.
		But the whole topic of the efficient packing of spheres is a crucial part of the mathematics that lies behind the error-detecting and error-correcting codes that are widely used to store information on compact disks and to compress information for efficient transmission around the world.
	www.maa.org /devlin/devlin_9_98.html (1088 words)

	Mathematical mysteries: Kepler's conjecture
		Kepler experimented with the problem and concluded that an arrangement known as the face centred cubic packing, a pattern favoured by fruit sellers, could not be bettered.
		It is also analogous to the problem of constructing optimal codes (see "Coding theory: the first 50 years" elsewhere in this issue).
		Although the sphere packing problem has frustrated mathematicians for nearly four centuries there is light at the end of the tunnel.
	pass.maths.org.uk /issue3/xfile (576 words)

	ON RANDOM PACKING OF SPHERES...
		By assuming the unit interval a car on the road, the problem is often called "random car parking problem".
		The most simple-minded algorithm which corresponds to the packing procedure stated above might be the following: the coordinates of the centre of a "test" sphere are determined by a random point which is uniformly distributed within the container.
		It is seen that, by using only the simple rejection scheme, it is almost hard to attain a complete packing when the size of the container is adequately large compared with the size of each sphere.
	www.mi.sanu.ac.yu /vismath/visbook/tanemura/index.html (1020 words)

	The Multicast Packing Problem
		multicast packing problem in which the network tries to accommodate simultaneously all the multicast groups while trying to avoid bottlenecks on the links for higher throughput (i.e., minimize the maximum link sharing among multicast groups).
		Minimization of maximum congestion is achieved at the expense of increasing the size of some multicast trees which in turn impacts the delay.
		The penalty term is a function of the amount of dilation from the size of the optimal tree obtained for each group multicast independently from the others (i.e., in isolation).
	www.comsoc.org /net/private/2000/jun/311_08net03-chen.html (762 words)

	bin packing problem (Site not responding. Last check: 2007-11-05)
		More formally, find a partition and assignment of a set of objects such that a constraint is satisfied or an objective function is minimized (or maximized).
		Note: A common form of the problem is, what is the least number of bins (containers of fixed volume) needed to hold a set of objects.
		Paul E. Black, "bin packing problem", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology.
	www.nist.gov /dads/HTML/binpacking.html (218 words)

	Set Packing and Partitioning (Site not responding. Last check: 2007-11-05)
		Other problems involving relationships between a set and some of its subsets are related to set covering.
		The set packing problem arises when each set element must appear in at most one subset.
		The set partitioning problem arises when each set element must appear in exactly one subset, and the constraints in this problem are equality constraints.
	mat.gsia.cmu.edu /orclass/integer/node9.html (383 words)

	Bin Packing Problem (Site not responding. Last check: 2007-11-05)
		The bin packing problem with no holes reduces immediately to asymmetric split.
		The traveling salesman problem, for example, has also received a great deal of attention, and there are algorithms that produce (nearly) the best solution for most "real world" graphs.
		Chip away at this tantalizing problem if you like, in your spare time, but remember, it has eluded the brightest minds of the past half century.
	www.mathreference.com /lan-cx-np,binp.html (422 words)

	Bin Packing Problem (Site not responding. Last check: 2007-11-05)
		Bin packing problem is NP complete when formulated as a decision problem.
		- Pack object i in bin j where j is the least index such that
		Packing generated by either FF or BF uses no more than
	www.cs.gsu.edu /~cscskp/Algorithms/NP/node11.html (131 words)

	1.6.9 Bin Packing (Site not responding. Last check: 2007-11-05)
		Suppose that you are manufacturing widgets with parts cut from sheet metal, or pants with parts cut from cloth.
		After our widgets have been successfully manufactured, we will be faced with another bin packing problem, namely how best to fit the boxes into trucks to minimize the number of trucks needed to ship everything.
		The best book available for this problem is Knapsack Problems : Algorithms and Computer Implementations by Silvano Martello and Paolo Toth.
	www.cs.sunysb.edu /~algorith/files/bin-packing.shtml (199 words)

	strip packing (Site not responding. Last check: 2007-11-05)
		Definition: Pack a set of rectangles into a strip of width 1 to minimize the height used.
		See also bin packing problem, knapsack problem, cutting stock problem.
		Paul E. Black, "strip packing", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology.
	www.nist.gov /dads/HTML/strippacking.html (129 words)

	Science News Online (8/15/98): Cracking Kepler's sphere-packing problem
		he familiar piles of neatly stacked oranges at a supermarket represent a practical solution to the problem of packing spheres as tightly as possible.
		Such an arrangement is known as face-centered cubic packing.
		In the 19th century, Carl Friedrich Gauss proved that face-centered cubic packing is the densest arrangement in which the centers of the spheres form a regular lattice.
	www.sciencenews.org /sn_arc98/8_15_98/fob7.htm (606 words)

	TBA ECP Ionisers Eliminate Static Packing Problem for Bemis Polyurethane Elastic Films
		However, it was found that the strips were becoming charged with static as they ran over the rollers and this was causing them to become tangled and stick to the plastic bags as they were being packed.
		TBA ECP's static eliminator bars are designed for use in a wide range of industries where static is a problem.
		Ion bars provide a non-contact solution to the problems caused by electrostatic fields within electronics EPAs (electrostatic protection areas), cleanrooms and other industrial applications where static is a problem.
	www.emediawire.com /releases/2006/6/prweb392654.php (1475 words)

	Packing Slip Problem - Peachtree Forums
		We receive a lot of request from customer to send copy of the packing slip signed.
		If peachtree have a link between the customer invoice and the peachtree database that we save the packing slip signed.
		However, you can print the Packing Slip from Peachtree, sign it, make a Xerox copy and file the copy in a folder.
	peachtreeusers.com /.forums/showthread.php?t=92 (102 words)

	Bin Packing, What is It? - Bin Packing - Developer Fusion: Connecting Developers Worldwide. C#, .NET, VB, Java, PHP and ...
		Bin Packing - Bin Packing, What is It?
		Would you know how or have a PHP version of this problem?
		Then in my code, where ever there is a reference to BinHeight, you would...
	www.developerfusion.co.uk /show/5540 (277 words)

	packing problem (Site not responding. Last check: 2007-11-05)
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	www.mbemineola.com /packing_problem.html (137 words)

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