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# Topic: Independent set

###### In the News (Sun 21 Jul 19)

 sciforums.com - Linear Transformations ,vn} a linearly independent set of vectors in R^n???? It is a SET of vectors that is or is not linearly independent. ,T(vn)} a linearly independent set of vectors in R^m? www.sciforums.com /showthread.php?t=29179   (2492 words)

 Algebraically independent - Wikipedia, the free encyclopedia In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In general, all the elements of an algebraically independent set over K are by necessity transcendental over K, but that is far from being a sufficient condition. Further, all these maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension. en.wikipedia.org /wiki/Algebraically_independent   (206 words)

 CS201: Assignments Define a graph whose vertices represent the set of possible code words, and add edges between any two code words sufficiently similar to be confused due to noise in the channel. The maximum independent set (the independent set having maximum number of vertices) of this graph defines the highest capacity code for the given communication channel. Let K be a subset of V. K is called an independent set if no two vertices in K are connected by an edge in E. The set K is called as the maximal (may not be maximum) independent set if further addition of any vertex destroys its independent set property. www.cse.iitk.ac.in /users/dsrkg/cs210old/html/assign10n.htm   (705 words)

 CSCI 3434 - NP Problems   (Site not responding. Last check: 2007-11-06) We can construct an independent set of size K in G as follows: For each clause in E, select any one literal that is true under the assignment A. There must be at least one such literal, for otherwise the clause (which is the "or" of its literals) would be false. Suppose that G has an independent set of size K. This set must have one node from each of the clauses, and it cannot have both an x-labelled node and a !x-labelled node. So, whenever a literal of the form x is in the independent set, we'll have our assignment A assign true to x, and whenever a literal of the form !x is in the independent set, then we'll have A assign false to x. www.cs.colorado.edu /~main/theory/s3-np.html   (1263 words)

 1.5.2 Independent Set   (Site not responding. Last check: 2007-11-06) Independent sets avoid conflicts between elements and hence arise often in coding theory and scheduling problems. Define a graph whose vertices represent the set of possible code words, and add edges between any two code words sufficiently similar to be confused due to noise. The maximum independent set of this graph defines the highest capacity code for the given communication channel. www.cs.sunysb.edu /~algorith/files/independent-set.shtml   (246 words)

 Challenging Benchmarks for Maximum Clique, Maximum Independent Set, Minimum Vertex Cover and Vertex Coloring - ...   (Site not responding. Last check: 2007-11-06) 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. To hide an independent set of size n in the instances of this graph model, we first select a vertex at random from each disjoint clique to form an independent set of size n, and then in the above process of generating random edges, no edge is allowed to violate this maximum independent set. 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)

 Independent Set   (Site not responding. Last check: 2007-11-06) The maximum independent set in G is exactly the maximum clique in G', so these problems are essentially identical. The independent set problem is in some sense dual to the graph matching problem. The maximum independent set of a tree can be found in linear time by   (1) stripping off the leaf nodes, (2) adding them to the independent set, (3) deleting the newly formed leaves, and then (4) repeating from the first step on the resulting tree until it is empty. www.cs.toronto.edu /~yuana/AlgorithmManual/BOOK/BOOK4/NODE173.HTM   (635 words)

 Hamiltonian Visibility Graphs in Sets of Independent Segments Theorem: For any set, S, of at least 2 independent segments, there exists a Hamiltonian Cycle, C, on the visibility graph of S such that every segment lying on the convex hull of S is included in C. Proof: Since s is not on the convex hull of S it must have segments on either side of it, so the sets L and R both contain at least 2 segments each. There are options to show the visibility graph of the set of segments and to show the convex hull of the set of segments. cgm.cs.mcgill.ca /~athens/cs507/Projects/2004/Joshua-Auerbach   (615 words)

 MetroActive | Sonoma Independent | Box Set With two prolific sources, Box Set has a wealth of new songs, which they have showcased in two early cassette-only recordings, followed by a pair of self-produced CDs, an eponymous debut in early 1994 and last year's 27. Even before recording 27, Box Set was singled out by Billboard a year ago for the industry magazine's "Honor Roll of America's Unsigned Talent," a slightly unnerving bit of recognition. Talks are under way with a small, East Coast independent label about a licensing agreement for a planned third CD, which is due to be recorded this spring. www.metroactive.com /papers/sonoma/01.04.96/box-set-9601.html   (711 words)

 Challenge Problems: Independent Sets in Graphs We may assume at least half the vertices in the independent set correspond to vectors with first coordinate 0 (since the graph is symmetric with respect to mapping vertices to their complements as 0-1 vectors). The size of the maximal independent set for length n and constant weight w is given in sequence A085684. The size of the maximal independent set for length n and constant weight w is given in sequence A085685. www.research.att.com /~njas/doc/graphs.html   (2953 words)

 Cliques, Independent Sets An independent set of G is an empty subgraph of G. maximal (maximum) independent set is defined is the same way as a maximal (maximum) clique. Note that an independent set of size k in the graph G is a clique of size k in the complement graph of G. The clique and independent set functions presented below are implemented using one of two algorithms, called here BranchAndBound and Dynamic. Finds a maximum independent set in the graph G. The maximum independent set is returned as a set of vertices. www.umich.edu /~gpcc/scs/magma/text1194.htm   (977 words)

 A Language-Independent Set of Benchmarks for Parallel Processors We present a set of six benchmarks which are language-independent, representative of mainstream scientiÞc computing, and have parallel content approaching 100% as parameters grow large. Even the Whetstone benchmark was originally intended as a weighted mean of several operations, with different sets of weights for different application mixes; yet the only numbers ever used are the default weights suggested by Curnow and Wichmann [2]. In general, however, a small set of short kernels can be representative of overall performance in scientiÞc applications; we make the tradeoff in the direction of simplicity because of the desire for language independence. www.scl.ameslab.gov /Publications/Gus/6-pack/6-pack.html   (4340 words)

 Approximations of Weighted Independent Set and Hereditary Subset Problems - us, Halld (ResearchIndex) Abstract: The focus of this study is to clarify the approximability of the important versions of the maximum independent set problem, and to apply, where possible, the technique to related hereditary subgraph and subset problem. In particular, we report improved performance ratios for the Weighted Independent Set problem in bounded-degree graphs, inductive graphs, and general graphs, and for the unweighted problem in sparse graphs. 9 Improved approximations of independent sets in bounded-degre.. citeseer.ist.psu.edu /241668.html   (727 words)

 Independent Set and Vertex Cover   (Site not responding. Last check: 2007-11-06) Both vertex cover and independent set are problems that revolve around finding special subsets of vertices, the first with representatives of every edge, the second with no edges. If S is the vertex cover of G, the remaining vertices S-V must form an independent set, for if there were an edge with both vertices in S-V, then S could not have been a vertex cover. This reduction shows that the hardness of vertex cover imples that independent set must also be hard. www.cs.toronto.edu /~yuana/AlgorithmManual/BOOK/BOOK3/NODE108.HTM   (320 words)

 maximal independent set   (Site not responding. Last check: 2007-11-06) Definition: A set of vertices in a graph such that for any pair of vertices, there is no edge between them and such that no more vertices can be added and it still be an independent set. Note: It is NP-complete to decide if there is maximal independent set of size k in a graph. Sandeep Kumar Shukla, "maximal independent set", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST. www.nist.gov /dads/HTML/maxmlndpndnt.html   (108 words)

 Articles - Basis (linear algebra)   (Site not responding. Last check: 2007-11-06) B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V, i.e., it is a generating set but no proper subset of B is. B is a maximal set of linearly independent vectors, i.e., it is a linearly independent set but no proper superset of B is. www.lastring.com /articles/Basis_(linear_algebra)?mySession=1c8d24802bdec06560ef3633f74e3b39   (1046 words)

 groebner::stronglyIndependentSets -- strongly independent set of variables   (Site not responding. Last check: 2007-11-06) is a set consisting of all maximal strongly independent sets of variables. A set is maximally strongly independent if no proper superset of it is strongly independent. Two maximally strongly independent set may be of different size. www.sciface.com /STATIC/DOC25/eng/groebner/strIndep.shtml   (169 words)

 Fish Data > INDEPENDENT DATA SETS   (Site not responding. Last check: 2007-11-06) We refer to these data as 'independent data sets' because they are not incorporated into the main StreamNet database. In order to post a new data set, two things are needed: 1) a description of the data set, and 2) the data set must be sent to us. The IDS submission tool is a very small program that fills these two needs by letting you provide the description of your own data set, and by sending your data set directly to StreamNet. www.streamnet.org /online-data/submitdatasets.html   (636 words)

 MEATHEAD RECORDS I guess this set represents a kind of "answer" to the music Industry's constant attempt to keep everything under its control - even though its easier and cheaper for artists to record and make CD's, the means of distribution is still out of reach. We were going to make 1000 sets, but since we were determined to keep the price as low as possible (9 CD's for the price of 2) and the set was so expensive to make, we could only afford to make 500. Each CD is housed in a specially designed fold-out which showcases really neat artwork that carries the theme of the set. www.angelfire.com /music/boxset   (545 words)

 Finding Large Independent Sets in Graphs and Hypergraphs Understanding the parallel complexity of this and related independent set problems on hypergraphs is a fundamental open issue in parallel computation. They also show that $\alpha_k(H)$ is the size of the maximum independent set for various hypergraph families. For graphs, we get an NC algorithm to find independent sets of size essentially that guaranteed by the general (degree-sequence based) version of Turán's theorem. epubs.siam.org /sam-bin/dbq/article/41973   (274 words)

 A Parallel Maximal Independent Set Algorithm   (Site not responding. Last check: 2007-11-06) The parallel construction of maximal independent sets is a useful building block for many algorithms in the computational sciences, including graph coloring and multigrid coarse grid creation on unstructured meshes. We present an efficient asynchronous maximal independent set algorithm for use on parallel computers, for well partitioned'' graphs, that arise from finite element (FE) models. For appropriately partitioned bounded degree graphs, it is shown that the running time of our algorithm under the P-RAM computational model is of $O(1)$, which is an improvement over the previous best P-RAM complexity for this class of graphs. sunsite.berkeley.edu /TechRepPages/CSD-98-993   (258 words)

 ICS 161, Winter 2001, Homework 5   (Site not responding. Last check: 2007-11-06) The following recursive algorithm takes as input an n-item array A, which we interpret as storing a complete binary tree in which the two children of A[i] are A[2i+1] and A[2i+2], the same interpretation we used in heapsort. In this tree, we say that a set of vertices is independent'' if it does not contain any parent-child pair. The algorithm returns the largest weight (sum of values) of any independent subset of the descendants of A[i]. www.ics.uci.edu /~eppstein/161/w01-hw5.html   (362 words)

 Irish News - Irish Independent Online - Sunday Independent , Irish newspapers, News Ireland   (Site not responding. Last check: 2007-11-06) AS NEWS filtered through of the former TD's death, the people of Lucan reacted with a mix of shock, sorrow and disbelief at the news. Download the daily Irish Independent onto your PDA or get the day's news direct to your mobile phone. Note: The Sunday Independent is now available Sunday mornings in Spain and Portugal. www.unison.ie /irish_independent   (394 words)

 Math 5718 Solutions 2 The set is linearly independent because it is not possible to find nonzero scalars Now we need to find another vector that can be expressed as a linear combination of the other three vectors. There are many other possible ways to extend this set to a basis. www-math.cudenver.edu /~wbriggs/5718s02/5718sol2/5718sol2.html   (438 words)

 A genetic algorithm-based heuristic for solving the weighted maximum independent set and some equivalent problems In this paper we present a genetic algorithm-based heuristic especially for the weighted maximum independent set problem (IS). We introduce several modifications to the basic genetic algorithm, by (i) using a crossover called two-fusion operator which creates two new different children and (ii) replacing the mutation operator by the heuristic-feasibility operator tailored specifically for the weighted independent set. The performance of our algorithm was evaluated on several randomly generated problem instances for the weighted independent set and on some instances of the DIMACS Workshop for the particular case: the unweighted maximum clique problem. www.palgrave-journals.com /cgi-taf/DynaPage.taf?file=/jors/journal/v48/n6/abs/2600405a.html   (190 words)

 Bush to set up independent intelligence inquiry: White House Until Friday, Bush had given a low-key reaction to the call foran independent probe into the reportedly US intelligence information failures in Iraq. Kay's call was echoed by the Democrats who have voiced strong support for the independent probe into the issue. The United States launched war against Iraq last March and toppled the regime of former Iraqi President Saddam Hussein on theground that Iraq had weapons of mass destruction and posed a threat to the United States. www.globalsecurity.org /intell/library/news/2004/intell-040202-pla-daily01.htm   (311 words)

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