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Topic: Packing and covering

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	PlanetMath: edge covering
		A minimal edge covering is simply an edge covering of the least possible.
		Cross-references: coverings, minimal, vertex, incident, edge, vertices, subset, graph
		This is version 5 of edge covering, born on 2002-05-26, modified 2002-07-06.
	planetmath.org /encyclopedia/EdgeCovering.html (79 words)

	The Geometry Junkyard: Covering and Packing
		There is only one parameter to optimize, the angle of the triangle to the lattice vectors; my answer is that the densest packing occurs when this angle is 15 or 45 degrees, shown below.
		Alon, Bóna, and Spencer show that one can't cover very much of an n by p(n) rectangle with staircase polyominoes (where p(n) is the number of these shapes).
		A candidate for the symmetric convex shape that is least able to pack the plane densely.
	www.ics.uci.edu /~eppstein/junkyard/cover.html (517 words)

	PENTAGON PACKING 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.
		For finite packing of pentagons, we can ask how to arrange a given number of non-overlapping equal pentagons in a circle so that the size of the pentagons will be a maximum.
		Conjectured solutions to the spherical pentagon packing problem without symmetry constraint have been established for up to n = 12, and with octahedral and icosahedral symmetry constraints for n = 24 and 72, respectively, were presented by Tarnai and Gáspár (2001).
	www.mi.sanu.ac.yu /vismath/proceedings/tarnai.htm (1329 words)

	TAU CS Colloquium --- Leah Epstein (Site not responding. Last check: 2007-09-20)
		We consider problems of resource allocation, these are various models of on-line and off-line bin packing and scheduling.
		We give a lower bound for on-line vector covering which is linear in the dimension.
		We discuss on-line machine covering of both identical and related machines and show that randomization decreases the competitive ratio for identical machines.
	www.math.tau.ac.il /~matias/98-FALL-TALKS/epstein.html (172 words)

	Packing Tape Covering - RC Groups
		I have covered my foamies with packing tape and this not only gives a great strength and stiffness but also if you paint it over the tape it does not absorbe any paint and so weights less.
		However it is difficult to cover the model without wrinkles, do you have any method to avoid these when covering double curvature surfaces.
		Covering tape works well on flat balsa control surfaces like the elevons but a round foam surface makes it difficult.
	www.rcgroups.com /forums/showthread.php?t=10957 (469 words)

	PACKING AND COVERING OF...
		On the other hand, the problem "How must the covering of a unit sphere be formed by N congruent spherical caps so that the angular radius of the spherical caps will be as small as possible?" is also important.
		Among the problems of packing and covering on the spherical surface, the Tammes problem is the most famous.
		Hereafter, we call the condition of Minkowski set of centers "Minkowski condition." If angular radii of spherical caps which cover the unit sphere under the Minkowski condition are concentrically reduced to half, the resulting spherical caps do not overlap.
	www.mi.sanu.ac.yu /vismath/visbook/sugimoto (1145 words)

	Research (Site not responding. Last check: 2007-09-20)
		My focus is to study functions associated with packing (covering) a given graph G with (by) specific types of graphs.
		Another useful application of packing and covering of graphs is with rigid graphs.
		I am seeing how these relate to packings and coverings of G; what types of graphs are usable for packing and covering for particular values of r (Examples : if r = 1, the graphs usable are trees; if r = 3/2, the graphs usable are 2-dimensional rigid graphs).
	www.math.tamu.edu /~kannan/research.html (426 words)

	Primal-dual approximation algorithms for a packing-covering pair of problems (Site not responding. Last check: 2007-09-20)
		The packing problem is a natural generalization of finding a (weighted) maximum independent set in an interval graph, the covering problem generalizes the problem of finding a (weighted) minimum clique cover in an interval graph.
		The problem pair involves weights and capacities; we consider the case of unit weights and the case of unit capacities.
		In each case we describe a simple algorithm that outputs a solution to the packing problem and to the covering problem that are within a factor of 2 of each other.
	www.edpsciences.org /articles/ro/abs/2002/01/ro2122/ro2122.html (194 words)

	Set Covering, Packing and Partitioning Problems (Site not responding. Last check: 2007-09-20)
		Exact Solution Approaches to the Set Covering, Packing and Partitioning Problems: Exact approaches to solving set partitioning, covering and packing problems require algorithms that generate both good lower and upper bounds on the true minimum value of the problem instance.
		One should note, however, that the set covering and packing problems are easier problems for heuristic search because for these problems, it is, in general, easy to find feasible solutions.
		As our understanding of the mathematical structure of the set partitioning, packing and covering polytopes improves, and with the continuing advancement in computer technology, it is likely that many difficult and important problems will be solved by being able to solve larger and larger set partitioning problems to proven optimality.
	iris.gmu.edu /~khoffman/papers/set_covering.html (2092 words)

	RANDOM SEQUENTIAL COVERING OF A SPHERE
		There are problems which deals with packing and covering of unit sphere by many identical spherical caps.
		These problems are applied to the globular histogenesis models of the biological organisms, to the uniform arrangement of observatories on the surface of the earth, to the information theories and so on.
		The center of the third spherical cap is chosen uniformly at random on the part of the perimeter of the first spherical cap which is not covered by the second spherical cap.
	members.tripod.com /vismath7/proceedings/sugimoto.htm (604 words)

	[No title] (Site not responding. Last check: 2007-09-20)
		The objective function in the covering and packing problems is to either minimize or maximize the number of sets that satisfy the constraints.
		The various problems we study are classified according to whether the constraints are all consecutive 1's or if there are also circular 1's constraints, and according to whether the constraints are all of covering type; all of packing type, or mixed.
		We present an $O(mn)$ time algorithm where all constraints are consecutive 1's of both packing and covering type.
	www.math.technion.ac.il /~techm/20031103123020031103lev (265 words)

	templeman's (Site not responding. Last check: 2007-09-20)
		Packing and covering problems are central problems in various branches of combinatorics, including Design Theory, Coding Theory, Graph Theory and Theoretical Computer Science.
		Recently, using some new combinatorial tools and using the probabilistic method, several mainstream packing and covering problems were solved.
		In this talk we survey some of these problems and their respective solutions, such as: covering graphs of bounded degree with a fixed tree, packing dense graphs with trees, packing and covering of complete graphs and orthogonal decompositions of complete graphs.
	www.cs.bgu.ac.il /~kojman/colloquium/yuster.html (96 words)

	Faster Approximation Algorithms for Packing and Covering Problems (Site not responding. Last check: 2007-09-20)
		In this talk we will discuss how to adapt a method proposed by Nesterov to design an algorithm that computes eps-optimal solutions to fractional packing problems by solving O(\sqrt{Kn}/eps) separable convex quadratic programs, where K denotes the maximum number of non-zeros per row and n denotes the the number of variables.
		For the special case of the maximum concurrent flow problem with rational capacities and demands the general algorithm can be modified to compute eps-optimal flow by solving shortest path problems -- the number of shortest paths that need to be computed grow as O(1/eps(\log(1/eps))) in eps, and polynomially in the size of the problem.
		Extensions to the maximum multicommodity flow problem, covering problems and mixed packing-covering problems will be described.
	www.stanford.edu /group/or/seminars/archives/iyengar_102004.html (224 words)

	Asymptotics of Packing, Covering, and Coloring Problems (Site not responding. Last check: 2007-09-20)
		The "incremental random" method is an approach to combinatorial problems which "constructs" an object of some desired type (e.g.
		packing, covering, coloring...) in small random increments, or more precisely, in increments which are shown to exist by random means.
		In recent years this approach has been used to establish good asymptotics for a number of combinatorial problems_e.g.
	www.siam.org /meetings/archives/dm96/kahn.htm (147 words)

	Fall 2004 COURSES
		These questions raised by Gauss, Minkowski, Hilbert, Helly, Konig, and many others turned out to be centrally important in number theory, coding theory, discrepancy theory, mathematical statistics, combinatorial optimization, and many areas of computer science from bin packing to robotics.
		This course offers an introduction to this rapidly developing field, where combinatorial and probabilistic (counting) methods play a crucial role.
		Topics: Geometry of numbers, Approximation of convex sets by polygons, Packing and covering with congruent convex discs, Lattice packing and lattice covering, The method of cell decomposition, Methods of Blichfeldt and Rogers, Efficient random arrangements, Epsilon-nets and transversals of hypergraphs, Geometric discrepancy, Bin packing.
	math.gc.cuny.edu /courses/fall_2004_courses_list.htm (291 words)

	Gerard Cornuejols (Site not responding. Last check: 2007-09-20)
		He is president of the INFORMS section on Optimization and committee chair of the new INFORMS Optimization prize.
		The integer programming models known as set packing and set covering have a wide range of applications, from pattern recognition to airline crew scheduling.
		In the last four, we cover decomposition results.
	www.win.tue.nl /wsk/onderwijs/2E941/courses/minicourses/cornuejols/MC-CCG-7.html (373 words)

	Finite Packing and Covering - Cambridge University Press (Site not responding. Last check: 2007-09-20)
		Connections to many other subjects were made, including crystallography, the local theory of Banach spaces, and combinatorial optimisation.
		This book, the first one dedicated solely to the subject, provides an in-depth state-of-the-art discussion of the theory of finite packings and coverings by convex bodies.
		Arrangements of congruent convex bodies in Euclidean space are discussed, and the density of finite packing and covering by balls in Euclidean, spherical and hyperbolic spaces is considered.
	www.cup.cam.ac.uk /catalogue/print.asp?isbn=0521801575&print=y (200 words)

	DC MetaData for: Methods in the Local Theory of Packing and Covering Lattices (Site not responding. Last check: 2007-09-20)
		Abstract: In this paper we are concerned with three lattice problems: the lattice packing problem, the lattice covering problem and the lattice packing-covering problem.
		For the lattice packing problem there are two classical algorithms going back to Minkowski and Voronoi.
		We describe several methods with examples to show that a lattice is a locally optimal solution to one of the three problems.
	www.math.uni-magdeburg.de /preprints/shadows/04-34report.html (214 words)

	Neal E. Young / publications (Site not responding. Last check: 2007-09-20)
		Sequential and parallel algorithms for mixed packing and covering
		The results generalize previous work on pure packing and covering (the special case when the constraints are all ``less-than'' or all ``greater-than'') by Michael Luby and Noam Nisan [
		Lagrangian-relaxation algorithms and greedy algorithms (in fact the greedy set-cover algorithm is an example), and are faster and simpler to implement than standard randomized-rounding algorithms.
	www.cs.ucr.edu /~neal/index.cgi?b=vita (4325 words)

	[No title] (Site not responding. Last check: 2007-09-20)
		Sequential and parallel algorithms for mixed packing and covering Neal Young Akamai Technologies and MIT Mixed packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients.
		Examples include multicommodity network flow, the Held-Karp lower bound on TSP, fractional relaxations of set cover, bin-packing, knapsack, scheduling problems, minimum-weight triangulation, etc. I'll describe approximation algorithms for the general class of problems.
		The results generalize previous work on pure packing and covering (the special case when the constraints are all ``less-than'' or all ``greater-than'') by Luby and Nisan [1993], Garg and Könemann [1998], and Lisa Fleischer [1999].
	www.cs.bu.edu /colloquium/LOG-03/03-05-2003-young.txt (134 words)

	abstract.html
		We study the problem of covering or packing a finite group with subgroups of a specified order and obtain bounds on the size of such covers and packings.
		Our main results provide characterizations of the elementary abelian groups by the existence of large packings or small covers, respectively.
		Hence large packings and small covers can always be thought of as geometric objects: they correspond to large partial t-spreads and small t-covers of a suitable projective space PG(d,p) for some prime p.
	www.math.mtu.edu /~tanner/Colloquia/Abstracts/jungnickel.html (96 words)

	Problem 56: Packing Unit Squares in a Simple Polygon
		BF01] conjecture the problem to be polynomially solvable.
		Optimal packing and covering in the plane are NP-complete.
		Approximation schemes for covering and packing problems in image processing and VLSI.
	maven.smith.edu /~orourke/TOPP/P56.html (164 words)

	Finite Packing and Covering - Cambridge University Press
		Finite Packing and Covering - Cambridge University Press
		Packing at Most 2d + 2 Balls on S
		Packings and Coverings with Respect to a Larger Ball
	www.cup.cam.ac.uk /catalogue/catalogue.asp?isbn=0521801575&ss=toc (139 words)

	Amazon.com: The Pursuit of Perfect Packing: Books: Tomaso Aste,Denis Weaire (Site not responding. Last check: 2007-09-20)
		The text is packed with both ordered and disorderly discussions of several packing problems that arise in daily life as well as in crystallography, physics and abstract mathematics.
		A good starting discussion for packing problems, even though several issues are not covered (look at Erich's Packing Center for the next edition).
		Chapters cover Circle packings, sphere packings, the Hales proof, seed shapes, honeycombs, bubbles, atoms and crystals, fractal aggregates, The Giant's Causeway, buckyballs, higher dimensional packings, and various odds and ends.
	www.amazon.com /exec/obidos/tg/detail/-/0750306483?v=glance (770 words)

	Set Packing and Partitioning (Site not responding. Last check: 2007-09-20)
		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.
		In fact, the group within American has spun off into a consulting operation (now part of Sabr Technologies), and is one of the most profitable areas for American Airlines.
	mat.gsia.cmu.edu /orclass/integer/node9.html (383 words)

	Szego - Constructive Characterizations for Packing and Covering by Trees (Site not responding. Last check: 2007-09-20)
		We also give a constructive characterization of graphs which have $k$ edge-disjoint spanning trees after {\em deleting} any edge of them.
		(These results can be found in A. Frank and L. ego: Constructive Characterizations for Packing and Covering by Trees, www.cs.elte.hu/~eszel/pl.html) P.S.: By a constructive characterization of a graph property, we mean a building procedure consisting of some simple steps so that the graphs obtained from a specified initial graph are precisely those having the property.
		For example, a graph is connected if and only if it can be obtained from a node by the operation: add a new edge connecting an existing node with either an existing node or a new one.
	www-leibniz.imag.fr /DMD/sem/2002/Szego.html (289 words)

	Prof (Site not responding. Last check: 2007-09-20)
		within UMass Lowell’s Department of Computer Science develops efficient geometric and combinatorial optimization algorithms, with an emphasis on algorithms for packing and covering problems.
		Efficient solutions to packing and covering problems are valuable in a wide variety of application areas, including manufacturing, graphics, visualization/data mining, and telecommunications.
		Lab research activities focus on: 1) identifying core problems whose solutions impact a collection of application areas; 2) developing efficient core algorithms to solve these problems; 3) developing application-specific algorithms based on core algorithms.
	www.cs.uml.edu /~kdaniels/LabPictures.htm (256 words)

	Weight Structure of Binary Codes and the Performance of Blind Search Algorithms (Site not responding. Last check: 2007-09-20)
		However, combinatorial coding theory is an independent area of investigations.
		Packing and covering radii are two important parameters of an algebraic code.
		In [1] two inequalities relating these parameters with thoroughness and sparsity of a blind search algorithm were established.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/sbrn/2000/0856/00/0856toc.xml&DOI=10.1109/SBRN.2000.889729 (215 words)

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