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Topic: Set cover problem

Related Topics

Vertex cover problem

FNP (complexity)

Clique problem

Hamiltonian path problem

Boolean satisfiability problem

List of computability and complexity topics

List of open problems in computer science

Traveling salesman problem

Andrey Ershov

Hamiltonian cycle

Hamiltonian path

PRoC

Newark, Arkansas

Robert Hayes

A World of His Own

	Set cover problem - Wikipedia, the free encyclopedia
		In the set cover decision problem, the input is a pair
		It is easy to see this by observing that an instance of set cover can be viewed as an arbitrary bipartite graph, with sets represented by vertices on the left, the universe represented by vertices on the right, and edges representing the inclusion of elements in sets.
		The set cover problem can be seen as a finite version of the notion of compactness in topology, where the elements of certain infinite families of sets can be covered by choosing only finitely many of them.
	en.wikipedia.org /wiki/Set_covering (692 words)

	Munagala, Kamesh; Babu; Shivnath; Motwani, Rajeev; Widom, Jennifer: The Pipelined Set Cover Problem
		Munagala, Kamesh; Babu; Shivnath; Motwani, Rajeev; Widom, Jennifer: The Pipelined Set Cover Problem
		We provide an abstraction of this problem as a generalization of set cover called pipelined set* cover, where the sets* are applied sequentially to the universe and the covered elements are discarded.
		We provide an abstraction of this problem as a generalization of set cover called pipelined set cover, where the sets are applied sequentially to the universe and the covered elements are discarded.
	dbpubs.stanford.edu /pub/2003-65 (579 words)

	Covering problem - Wikipedia, the free encyclopedia
		In combinatorics and computer science, the covering problem is a type of general question: if a certain structure covers another, or how many structures are required to cover another?
		For Petri nets, for example, the covering problem is defined as the question if for a given marking, there exists a run of the net, such that some larger (or equal) marking can be reached.
		See also: set cover problem, vertex cover problem, packing problem.
	en.wikipedia.org /wiki/Covering_problem (135 words)

	[No title]
		An implication table is made, then a cover function is formed and simplified, and the term(s) of the simplified cover function with the minimal set factors is(are) optimal.
		This is a variant of the "set cover" problem, which is NP-complete in the general case.
		By considering which of the three- and four-element target sets are formed as unions, we find a number of ten-set solutions and a nine-set solution: 8,67,78, 15,26,37,48, 2568,3567 where the nonprimitive sets are realized as 158=15+8, 268=26+8, 478=48+78, 1578=15+78, 367=37+67, 4678=48+67, and 24568=48+2568.
	www.math.niu.edu /~rusin/known-math/95/weaving (529 words)

	Top: SetCoverByGreedy
		The partial set cover problem is, given a collection of sets and an integer K, to find a minimum number of sets covering all but some K of the elements.
		The Set Cover problem is, given a collection of sets, to choose a minimum number of those sets so that every element is in at least one of the chosen sets.
		The weighted set cover problem is, given a collection of sets each with a non-negative weight, to choose a set cover minimizing the total weight of the chosen sets.
	www.cs.ucr.edu /~neal/wiki/wiki.pl?action=browse&diff=1&id=SetCoverByGreedy (702 words)

	Vertex Cover
		A set of vertices defines a vertex cover in graph G iff the corresponding subsets define a set cover in the given instance.
		The simplest heuristic for vertex cover selects the vertex with highest degree, adds it to the cover, deletes all adjacent edges, and then repeats until the graph is empty.
		A problem that might seem closely related to vertex cover is edge cover, which seeks the smallest set of edges such that each vertex is included in one of the edges.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE174.HTM (838 words)

	CS 510 Homework 2 Solutions (Site not responding. Last check: 2007-11-02)
		Prove that set cover is NP-complete with a reduction from vertex cover.
		Input: a set X of n elements, a family F of subsets of X, and an integer k.
		Thus C' is a set cover for X and F with size k.
	max.cs.kzoo.edu /Algs/CS510S98/hw2Soln.html (573 words)

	Approximating Art Galleries
		reduce the art gallery problem to set cover and approximate the latter one.
		The reduction of the art gallery to the set cover problem is randomized and can be stated as: 1.
		Approximate the set cover problem, i.e., find a minimal set of candidate positions from which the entire polygon is visible.
	www.ais.fraunhofer.de /ARC/3D/download/icar2003/node4.html (164 words)

	ECCC Report TR97-004 and related Papers (Site not responding. Last check: 2007-11-02)
		An instance of the set cover problem with $m$ sets is dense if there is $\epsilon>0$ such that any element belongs to at least $\epsilon m$ sets.
		We show that the dense set cover problem can be approximated with the performance ratio $c\log n$ for any $c>0$ and it is unlikely to be NP-hard.
		Though this problem is shown to be still MAX-SNP-hard as in general graphs, we find a better approximation algorithm with the performance ratio $2\over{1+\epsilon}$.
	www.eccc.uni-trier.de /eccc-reports/1997/TR97-004 (210 words)

	Set Cover
		It is instructive to model vertex cover as an instance of set cover.
		Although vertex cover is just a set cover problem in disguise, you should take advantage of the fact that better algorithms exist for vertex cover.
		Hitting set is, in fact, dual to set cover, meaning it is exactly the same problem in disguise.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK5/NODE201.HTM (1046 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		We provide an abstraction of this problem as a generalization of set cover called pipelined set cover, where the sets are applied sequentially to the elements to be covered and the elements covered at each stage are discarded.
		We show that several natural heuristics for this NP-hard problem, such as the greedy set-cover heuristic and a local-search heuristic, can be analyzed using a linear-programming framework which bounds not only the approximation ratio, but also the running time of the corresponding algorithms.
		We use this framework to show that the greedy and local-search algorithms are 4-approximations for pipelined set cover.
	www.icdt2005.inf.ed.ac.uk /Acc/abstracts/mbmw.html (221 words)

	Neal E. Young / publications (Site not responding. Last check: 2007-11-02)
		The algorithm reduces the problem to the minimum-weight balanced-separator problem; the performance guarantee of the algorithm is within a factor of 4 of the performance guarantee of the balanced-separator procedure.
		For the second set of results, the paper presents empirical studies of paging algorithms, documenting that in practice, on ``typical'' cache sizes and sequences, the performance of paging strategies are much better than their worst-case analyses in the standard model suggest.
		The minimum-balance problem is to find a ``weighting'' of the vertices so that adjusting the edge costs by the vertex weights yields a graph in which, for every cut, the minimum weight of any edge crossing the cut in one direction equals the minimum weight of any edge crossing the cut in the other direction.
	www.cs.ucr.edu /~neal/index.cgi?b=vita (4286 words)

	1.5.3 Vertex Cover (Site not responding. Last check: 2007-11-02)
		Excerpt from The Algorithm Design Manual: Vertex cover is a special case of the more general set cover problem, which takes as input an arbitrary collection of subsets S = (S_1, \ldots, S_n) of the universal set U = \{1,\ldots,m\}.
		To turn vertex cover into a set cover problem, let U be the complete set of edges, and create $S_i$ to be the set of edges incident on vertex i.
		A set of vertices defines a vertex cover in graph $G$ iff the correspondinag subsets define a set cover in the given instance.
	www.cs.sunysb.edu /~algorith/files/vertex-cover.shtml (222 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		We assume that the elements and sets are known in advance to the algorithm, however, it is not known in advance which elements will be given by the adversary to the algorithm.
		(In general, the set of elements given by the adversary may be a strict subset of the total set of elements.) The objective is to minimize the total cost of the sets chosen by the algorithm.
		The performance of the algorithm is the ratio between the cost the cover that it produces and the optimal cost of covering the elements given by the adversary.
	www.math.technion.ac.il /~techm/20031113121520031113nao (291 words)

	Room Reservation PSet
		This problem set is different in style from the previous one, and you should work on it differently.
		Although there are no formal checkpoints required for this problem set, it would be a good idea to show your work to a TA before continuing with exercise 3, to make sure your data model is adequate for completing the problem set.
		This problem set was written by Philip Greenspun and Hal Abelson in February 1999 for MIT Course 6.916.
	philip.greenspun.com /teaching/psets/rooms/tcl/rooms.adp (5007 words)

	Feature Article
		Given a set of mechanisms that are known or thought to be in use, we reverse the cause-effect model to produce a ranking of all causes that could be associated with each of the observed effects.
		The covering analysis indicates that these methods have the potential of detecting all of the identified mechanisms, indicates that the defenses with the best coverage are (1) time, location, function, and other similar access limitations, (2) redundancy, and (3) filtering devices and that in combination, these cover almost all of the identified attack mechanisms.
		Defense postures (i.e., sets of defensive mechanisms that can be placed in the network) can also be analyzed for their impact on cost (or probability) to increase the minimum cost (or maximum probability) of attack within a fixed budget for defense or minimize the cost (or probability) of successful defense for a given attack budget.
	www.all.net /journal/ntb/cause-and-effect.html (15812 words)

	Placing Servers in Overlay Networks
		An instance of the set cover problem is that given a base set of elements and a family of sets that are subsets of this base set, find the minimum number of sets such that their union includes all elements in the base set.
		By solving the set cover problem, we find the minimum number of servers and their locations, that will cover all clients within the service range.
		The rest of the chapter is organized as follows: we introduce the formal problem definitions and the algorithms in Section 5.1; we then describe the network models used in our simulations in Section 5.2; Section 5.3 presents simulation results; Section 5.4 discusses some of the related work; and in Section 5.5, we summarize our results.
	www.arl.wustl.edu /~sherlia/thesis/chap5/node1.html (998 words)

	1.7.1 Set Cover (Site not responding. Last check: 2007-11-02)
		Excerpt from The Algorithm Design Manual: Set cover arises when you try to efficiently acquire or represent items that have been packaged in a fixed set of lots.
		An interesting application of set cover is Boolean logic minimization.
		Given a set of feasible ``and'' terms, each of which covers a subset of the vectors we need, we seek to ``or'' together the smallest number of terms that realize the function.
	www.cs.sunysb.edu /~algorith/files/set-cover.shtml (253 words)

	set cover (Site not responding. Last check: 2007-11-02)
		Definition: A set of sets whose union has all members of the union of all sets.
		The set cover problem is to find a minimum size set.
		Paul E. Black, "set cover", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/setcover.html (93 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		We present an $O(V^2 +H)$ time, $\B$-approximation algorithm for this problem, where $\B \geq 2$ is an upper bound on the edge cardinality of the hypergraph, and $H$ is the size of the hypergraph (i.e.
		A set $C \subseteq E$ is called a {\em clique-complement cover} (or just a {\em cover}) if the set of edges $E-C$ induces a complete graph.
		The Min Clique-Complement problem is: given $G=(V,E)$, and for every edge $e \in E$ a weight $\omega(e) \geq 0$, find a cover with a minimum total weight.
	www.cs.technion.ac.il /~reuven/SAVE/PAPERS/LOCAL_RATIO/APROX98/summery (250 words)

	On the Red-Blue Set Cover Problem - Carr, Doddi, Konjevod, Marathe (ResearchIndex)
		Abstract: Given a finite set of "red" elements R, a finite set of "blue" elements B and a family S ` 2 R[B, the red-blue set cover problem is to find a subfamily C ` S which covers all blue elements, but which covers the minimum possible number of red elements.
		We note that Red-Blue Set Cover is closely related to several combinatorial optimization problems studied earlier.
		8 Primal-dual RNC approximation algorithms for set covering an..
	citeseer.ist.psu.edu /260486.html (598 words)

	ipedia.com: NP-complete Article (Site not responding. Last check: 2007-11-02)
		In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solv...
		One example of an NP-complete problem is the subset sum problem which is: given a finite set of integers, determine whether any non-empty subset of them adds up to zero.
		Not all NP-complete problems have good approximation algorithms, and for some problems finding a good approximation algorithm is enough to solve the problem itself.
	www.ipedia.com /np_complete_1.html (1137 words)

	Brian's Digest: Set Covering Problem
		For the solution of the problem of Patrizio you need the minimal number of workers so that for each job permissible solution of the corresponding marriage problem exists.
		It is not nessesary to generate such a solution because the marriage theorem gives sufficient conditions for the existance of such a solution.
		It is easy to show a reduction from the vertex-coloring problem to this problem.
	www.worms.ms.unimelb.edu.au /digest/set_covering.html (1420 words)

	[No title]
		Sum of values on sets covering it is at least 1/k'.
		p-center problem: Given graph G and number p, find set of p "center" vertices (firehouses) such that sets of radius R from these vertices cover all of G, for as small a value of R as possible.
		Again there is some y that covers B but B doesn't cover y so some center C != B covers y.
	www.cs.cmu.edu /People/avrim/Approx00/lectures/lect0124 (857 words)

	[No title]
		For the general vertex cover problem, Nemhauser and Trotter \cite{NemTro75} developed a local optima algorithm that implies a 2-approximation.
		The {\em set-cover} problem is a generalization of the vertex cover problem.
		A set $C \subseteq V$ is called a {\em set cover} of a hypergraph $H=(V,E)$ if every (hyper)edge has at least one endpoint in $C$, i.e., $\forall_{e \in E}\ e \cap C \neq \phi$.
	www.cs.technion.ac.il /~reuven/SAVE/PAPERS/PARTIAL_COVER/OLD/SODA/save1 (1286 words)

	MS&E 130/231: Information Systems, Winter 2003-04
		all sets in the collection have the same cost): First, solve the fractional relaxation of the set cover problem optimally.
		Explain how that proof can be used as a fl-box to obtain an O(log n) approximation guarantee for the greedy algorithm for unweighted set cover.
		Consider the maximum independent set problem: given a graph G = (V,E), find the largest subset S of V such that there is no edge between any two vertices in S.
	www.stanford.edu /~ashishg/approx/hand2-hw1.html (345 words)

	Campus Event Calendar: Naveen Sivadasan (11/24/2003 in 46.1 - MPII/024)
		Once a new element is given, the algorithm has to cover it by some set of S containing it.
		We assume that the elements of X and the members of S are known in advance to the algorithm, however, the set X' \subset X of elements given by the adversary is not known in advance to the algorithm.
		Let C denote the family of sets in S that the algorithm chooses.
	domino.mpi-sb.mpg.de /internet/events.nsf/0313ea5028c9be24c125642a0009eca7/5de280f70792b87ec1256ddd004d3022!OpenDocument&Date=2003-11-24 (320 words)

	On the Greedy Algorithm for a Covering Problem (Site not responding. Last check: 2007-11-02)
		On the Greedy Algorithm for a Covering Problem
		The k-set-cover problem is an abstraction of many commonly-arising combinatorial problems, such as, for instance, certain problems in facilities location and task assignment; however, it is NP-hard.
		Also discussed is a specific example of the k-set-cover problem, namely, covering points in the plane with k axes-parallel rectangles.
	www.cs.umn.edu /tech_reports_upload/1993/TR_93-13_On_the_Greedy_Algorithm_for_a_Covering_Problem.html (107 words)

	[No title]
		Suppose that A and B are problems in P, C and D are in NP, E is NP-complete.
		For a problem X, XC refers to the complement problem of X. Explain your answer in one sentence.
		Instance: Set S and a collection C of subsets of S, and integer k Question: Does there exist a subset D of C such that D covers S and D M with
	www.cse.msu.edu /~cse860/exam1.doc (596 words)

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