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Topic: Vertex cover problem

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	Vertex cover problem - Wikipedia, the free encyclopedia
		In computer science, the vertex cover problem or node cover problem is an NP-complete problem in complexity theory, and was one of Karp's 21 NP-complete problems.
		Vertex cover is NP-complete, which means it is unlikely that there is an efficient algorithm to solve it.
		Vertex cover is closely related to Independent Set problem by this theorem: a graph with n vertices has a vertex cover of size k if and only if the graph has an independent set of size n − k.
	en.wikipedia.org /wiki/Vertex_cover_problem (390 words)

	vertex cover problem (Site not responding. Last check: 2007-10-22)
		A vertex cover of an undirected graph is a subset of the vertices of the graph which contains at least one of the two endpoints of each edge:
		A brute force algorithm to find a vertex cover in a graph is to list all subsets of vertices of size and check each one to see whether it forms a vertex cover.
		This algorithm is exponential in, but not in the size of the graph, i.e., vertex cover is fixed-parameter tractable with respect to.
	www.yourencyclopedia.net /vertex_cover_problem.html (291 words)

	Fixed-parameter algorithm - Wikipedia, the free encyclopedia
		It is believed to be difficult to find efficient, exact, deterministic solutions to these problems; all known solutions require time that is exponential in the total size of the inputs.
		For instance, for the vertex cover problem, the parameter can be the number of vertices in the cover.
		This proves that vertex cover is fixed-parameter tractable with respect to this parameter.
	en.wikipedia.org /wiki/FPT (308 words)

	Vertex cover problem - Wikipedia, the free encyclopedia
		A vertex cover of an undirected graph $G = (V, E) is a subset V' of the vertices of the graph which contains at least one of the two endpoints of each edge:$
		A brute force algorithm to find a vertex cover in a graph is to list all subsets of vertices of size $k and check each one to see whether it forms a$ vertex cover.
		This algorithm is exponential in $k, but not in the size of the graph, i.e.,$ vertex cover is fixed-parameter tractable with respect to $k .$
	www.encyclopedia-online.info /Vertex_cover_problem (297 words)

	Note_Oct14 (Site not responding. Last check: 2007-10-22)
		Definition: Given a graph G(V, E), a vertex cover S is a subset of V, that covers all the edges in E. The size of a vertex cover is the number of vertices in it.
		If we remove the vertex cover and all the edges covered by it from the graph, there will be no edges left between any pair of the remaining vertices; otherwise, such an edge is not covered by any vertex in the vertex cover.
		Suppose we could solve the vertex cover problem in polynomial time and we know the minimum size in all the vertex covers for a given graph is k, we would like to find such a vertex cover of size k.
	www.msci.memphis.edu /~giri/7713/f98/wyang/wyang.html~ (1666 words)

	Station Information - Vertex cover problem
		A vertex cover in a graph is a subset of the verticies of the graph, chosen with the property that one of the endpoints of each edge is in the subset.
		The problem is a decision problem, so we wonder if a vertex cover of size k exists in the graph.
		A brute force algorithm to find a vertex cover in a graph is to list all subsets of vertices, V and check each one to see if it forms a vertex cover.
	www.stationinformation.com /encyclopedia/v/ve/vertex_cover_problem.html (152 words)

	Vertex cover problem (Site not responding. Last check: 2007-10-22)
		Vertex cover is NP-complete, which means it is unlikely that there is anefficient algorithm to solve it.
		A brute force algorithm to find a vertex cover in a graph is to list allsubsets of vertices of size k and check each one to see whether it forms a vertex cover.
		Thisalgorithm is exponential in k, but not in the size of the graph, i.e., vertex cover is fixed-parameter tractable with respect tok.
	www.therfcc.org /vertex-cover-problem-194556.html (254 words)

	CS523: Growing practice problem set 2 (Site not responding. Last check: 2007-10-22)
		Clique problem: Given a graph G and a positive integet k, the clique problem asks whether a clique of size k exists in G. Show that the clique problem is NP-hard by reducing the 3-CNF-SAT problem to the clique problem.
		Vertex cover: A vertex cover of an undirected graph G = (V, E) is a subset V' of V such if (u, v) belongs to E, then at least one of u and v belong to V'.
		Vertex cover problem: Given a graph G and a positive integet k, the vertex cover problem asks whether a vertex cover of size k exists in G. Show that the vertex cover problem is NP-hard by reducing the clique problem to the vertex cover problem.
	web.engr.oregonstate.edu /~saurabh/cs523/practice3.html (701 words)

	Covering all bases - for printer (Site not responding. Last check: 2007-10-22)
		The problem comprises the 'size of the city' (the number of vertices, assumed to be connected at random), the connectivity (number of streets meeting at each vertex) and the fraction of occupied vertices.
		The vertex cover problem belongs to a class called 'NP-complete' problems, which are the hardest to solve.
		In the vertex cover problem, for instance, it is relatively easy to find a solution if there are almost as many police as intersections, and relatively easy to see that cover is impossible if the police are vastly outnumbered.
	www.nature.com /nsu_new/nsu_pf/000706/000706-1.html (719 words)

	Computational complexity theory - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-22)
		A single "problem" is an entire set of related questions, where each question is a finite-length string.
		The time complexity of a problem is the number of steps that it takes to solve an instance of the problem, as a function of the size of the input, (usually measured in bits) using the most efficient algorithm.
		A decision problem is a problem where the answer is always YES/NO. For example, the problem IS-PRIME is: given an integer written in binary, return whether it is a prime number or not.
	encyclopedia.learnthis.info /c/co/computational_complexity_theory.html (922 words)

	NP-Complete Problems (Site not responding. Last check: 2007-10-22)
		Vertex Cover Problem: A vertex cover of an undirected graph G = (V, E) is a subset V' C V such that if (u, v) belongs to E, then u belongs to V' or v belongs to V' (or both).
		Vertex Cover is a set of vertices that touches all the edges of G. Given an undirected graph G = (V, E), we define the complement of G as
		The output of the reduction algorithm is the instance of the vertex cover problem.
	www.msci.memphis.edu /~giri/7713/f99/kolli/NPComplete.html (922 words)

	Department of Computer Science
		In this paper we study the performance ratio of the greedy cover algorithm for this "partial vertex cover" problem and compare it to random choice and to optimal complete fractional cover.
		He proved that the performance ratio of the greedy cover algorithm for the partial vertex cover problem for weighted hypergraphs is <= 2 ln(m) regardless of epsilon >= 0.
		As a corollary, the greedy cover algorithm has a performance ratio 1 / epsilon in the unweighted case if the hypergraph is regular and uniform.
	www.cs.uchicago.edu /events/274 (329 words)

	Vertex cover problem -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		Vertex cover is (Click link for more info and facts about NP-complete) NP-complete, which means it is unlikely that there is an efficient algorithm to solve it.
		Vertex cover is closely related to (Click link for more info and facts about Independent Set problem) Independent Set problem by the theorem - a graph with vertices has a vertex cover of size if and only if the graph has an independent set of size.
		A (Click link for more info and facts about brute force) brute force algorithm to find a vertex cover in a graph is to list all subsets of vertices of size and check each one to see whether it forms a vertex cover.
	www.absoluteastronomy.com /encyclopedia/v/ve/vertex_cover_problem.htm (459 words)

	1.5.3 Vertex Cover (Site not responding. Last check: 2007-10-22)
		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)

	CSCI 3650 Answers to practice questions for exam 3
		Is it known whether the vertex cover problem is in the class P? Is it known whether the vertex cover problem is in the class NP?
		The vertex cover problem is known to be NP-complete.
		Take the set S of all vertices that are not in I. Then S must be a vertex cover of G. Any edge not covered by S would have to be between two vertices that are not in S. But there are no such edges, since I is an independent set.
	www.cs.ecu.edu /~karl/3650/sum01/exam3soln.html (844 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Now if a vertex cover is found in this multigraph in polynomial time then a vertex cover in the general graph may also be found in polynomial time, which is impossible.
		For any vertex cover problem the previous reduction may be used to create an instance of the given problem.
		Any hamiltonian circuit problem in a $3$ degree regular graph may be reduced to an instance of a eularian cycle problem.
	math.ucsd.edu /~fan/math188/key5.txt (553 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		The problem has applications in VLSI mask generation, incremental update of raster displays, and image compression.
		Covering with similar rectangles (having a given x/y ratio) is an equivalent problem.", } @Article{BarGot96, author = "R. Bar-Yehuda C. Gotsman", title = "Time/Space Tradeoffs for Polygon Mesh Rendering", journal = " ACM Trans.
		For general vertex weights, the performance ratio becomes $\min\{2\Delta^2, 4 \log_2 n\}$ where $\Delta$ denotes the maximum degree in $G$.
	www.cs.technion.ac.il /~reuven/CV/bibtex/BB (1887 words)

	Citations: Ramsey numbers and an approximation algorithm for the vertex cover problem - Monien, Speckenmeyer ... (Site not responding. Last check: 2007-10-22)
		The parameterized complexity [13] of UVC recently has received considerable interest [6, 10, 13, 15, 26, 32] Here, for a given k, the question is to find a vertex cover of at most k vertices or to report no if no vertex cover of size at most k exists.
		....greedy algorithm is to pick any edge from the graph, put both endpoints in the vertex cover, and delete these endpoints together with their incident edges from the graph.
		However, unless P = NP, Vertex Cover has no polynomial time approximation scheme [2] and it is known to be not approximable to a ratio 0.1666 [17] Although Vertex Cover is hard to approximate, it has turned out that it is easy to parameterize : Vertex Cover has seen quite some history of....
	citeseer.ist.psu.edu /context/9616/0 (3313 words)

	CSCI 3650 Practice questions for exam 2
		A vertex cover of an undirected graph G is a set S of vertices such every edge in G has at least one of its endpoints in S.
		The vertex cover problem is: Given an undirected graph G, find a vertex cover of G that has the smallest possible size.
		A vertex cover needs to cover all of the edges, where an edge is covered by choosing a vertex from that edge.
	www.cs.ecu.edu /~karl/3650/sum04/practice2.html (671 words)

	Computational complexity theory Article, Computationalcomplexitytheory Information (Site not responding. Last check: 2007-10-22)
		For example, the problem FACTORIZE is: given an integer written in binary, return all of the prime factors of that number.A particular question is called an instance.
		The time complexity of a problem is the number of steps that it takes to solve an instance of the problem, asa function of the size of the input, (usually measured in bits) using themost efficient algorithm.
		A decisionproblem is a problem where the answer is always YES/NO. For example, the problem IS-PRIME is: given an integerwritten in binary, return whether it is a prime number or not.
	www.anoca.org /problem/problems/computational_complexity_theory.html (909 words)

	mikero.com - Perl Regular Expression Matching is NP-Hard (Site not responding. Last check: 2007-10-22)
		"cover" their incident edges, and we want all edges covered by some vertex.
		, indicating that vertices 1, 2, and 4 are a 3-vertex cover.
		If there were an efficient (polynomial-time) algorithm for computing whether or not a regex matched a certain string, we could use it to quickly compute solutions to the graph 3-colorability problem, and, by extension, to the knapsack problem, the travelling salesman problem, etc. etc.
	www.mikero.com /code/vertex-cover2.html (161 words)

	Set Cover (Site not responding. Last check: 2007-10-22)
		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.
		Since the vertex cover problem is NP-complete, the set cover problem must be at least as hard.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK5/NODE201.HTM (1046 words)

	Challenging Benchmarks for Maximum Clique, Maximum Independent Set, Minimum Vertex Cover and Vertex Coloring - ... (Site not responding. Last check: 2007-10-22)
		A vertex cover is a subset S of vertices such that each edge of G has at least one of its endpoints in S. It is easy to see that given an independent set of a graph, all vertices not in the set form a vertex cover.
		Finding challenging benchmarks for the maximum independent set problem (or equivalently, the minimum vertex cover problem) is not only of significance for experimentally evaluating the algorithms of solving this problem but also of interest to the theoretical computer science community.
		Since a clique is an independent set in the complementary graph, the maximum independent set problem and the maximum clique problem (which is one of the first shown to be NP-hard and has been extensively studied in graph theory and combinatorial optimization) are essentially equivalent.
	www.nlsde.buaa.edu.cn /~kexu/benchmarks/graph-benchmarks.htm (1556 words)

	Sample Proof of NPC
		This particular proof was chosen because it reduces 3SAT to VERTEX COVER and involves the transformation of a boolean formula to something geometrical.
		Question: Is there a vertex cover of size K or less for G, i.e., a subset V' of V with the size of V' less than K such that every edge has at least one endpoint in V'.
		The other 2m vertices (or less) must be covering the triangles since each triangle requires two vertices to be covered.
	cgm.cs.mcgill.ca /~cwu25/proj507/npproof.html (689 words)

	Abstract for Asaf Levin's talk (Site not responding. Last check: 2007-10-22)
		The cost of including a vertex v in the solution is c(v).
		The minimum generalized vertex cover problem is to compute a minimum cost set of vertices.
		We study the complexity of the problem when the costs $d_0(e)=1$, $d_1(e)=\alpha$ and $d_2(e)=0$ $\forall e \in E$ and $c(v)=\beta$ $\forall v \in V$ for all possible values of $\alpha$ and $\beta$.
	www.cs.berkeley.edu /~elitza/Seminars/Abstracts/10-14-02.html (139 words)

	DRUM: Item 1903/1272 (Site not responding. Last check: 2007-10-22)
		In this paper we study the capacitated vertex cover problem, a generalization of the well-known vertex cover problem.
		Given a graph $G=(V,E)$, the goal is to cover all the edges by picking a minimum cover using the vertices.
		The problem is clearly NP-hard as it generalizes the well-known vertex cover problem.
	hdl.handle.net /1903/1272 (217 words)

	Vertex cover problem - Encyclopedia, History, Geography and Biography
		Vertex cover problem - Encyclopedia, History, Geography and Biography
		Vertex cover is closely related to Independent Set problem by this theorem: a graph with $n$ vertices has a vertex cover of size $k$ if and only if the graph has an independent set of size $n-k$ .
		Vertex cover problem, See also and Further reading.
	www.arikah.net /encyclopedia/Vertex_cover_problem (427 words)

	PTAS for uncovered by min vertex cover problem? (Site not responding. Last check: 2007-10-22)
		I that there is no Polynomial Time Approximation Schemes(PTAS) for the minimum size of a vertex cover of a graph.
		However, I am trying to find out whether there is a PTAS for the number of vertcies uncovered by the minimum vertex cover.
		It is easy to calculate one from the other - if 90 verticies in a 100 vertex graph are in the minimum cover then there remain 10 verticies uncovered.
	www.seriousliving.net /new-51645-13.html (152 words)

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