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# Topic: Clique problem

###### In the News (Tue 21 Oct 14)

 Clique problem - Wikipedia, the free encyclopedia A clique in a graph is a set of pairwise adjacent vertices, or in other words, an induced subgraph which is a complete graph. Then, the clique problem is the problem of determining whether a graph contains a clique of at least a given size k. The clique problem's NP-completeness follows trivially from the NP-completeness of the independent set problem, because there is a clique of size at least k if and only if there is an independent set of size at least k in the complement graph. en.wikipedia.org /wiki/Clique_problem   (407 words)

 Clique - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-05) With this, the social role of the "outcast" is defined, as individuals that the queen bee dislikes may be classified by her as such, thus encouraging clique members to victimize the outcast, in order to continue to be part of the clique or to receive praise from the queen bee. Clique members may be influenced through peer pressure to engage in actions perceived by most as negative or damaging, such as tobacco smoking or drug use. Cliques may also be a source of distraction from studies, both for clique members and for the outcasts they victimize; outcasts may suffer long-term psychological damage resulting from the bullying they suffer. www.sciencedaily.com /encyclopedia/clique   (812 words)

 CS523: Growing practice problem set 2   (Site not responding. Last check: 2007-11-05) Clique: A clique in an undirected graph G = (V, E) is a subset V' of V, each pair of which is connected by an edge in E. The size of a clique is the number of vertices it contains. 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 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)

 Clique (graph theory) - Wikipedia, the free encyclopedia In graph theory, a clique in an undirected graph G, is a set of vertices V such that for every two vertices in V, there exists an edge connecting the two. Finding the largest clique in any graph (the clique problem) for example, is NP-complete. The opposite of a clique is an independent set, in the sense that every clique corresponds to an independent set in the complement graph. en.wikipedia.org /wiki/Clique_(graph_theory)   (168 words)

 Maximum Clique Problem   (Site not responding. Last check: 2007-11-05) This problem was the Maximal Clique Problem: given a group of vertices some of which have edges in between them, the maximal clique is the largest subset of vertices in which each point is directly connected to every other vertex in the subset. This is a problem because, as noted early, the number of cliques grows exponentially. Each possible clique was represented by a binary number of N bits where each bit in the number represented a particular vertex. www.stanford.edu /~alexli/soco/clique.htm   (770 words)

 Clique   (Site not responding. Last check: 2007-11-05) In graph theory, a clique in a undirected graph G, is a set of vertices V' such that for every two vertices in V, there exists an edge connecting the two. The clique problem refers to the problem of finding the largest clique in any graph G. This problem is NP-complete, and as such, many consider that it is unlikely that an efficient algorithm for finding the largest clique of a graph exists. If we already know that the independent set problem is NP-complete, then it is easy to prove, as the size of the largest clique is the same as the size of the largest independent set in the inverse graph. www.serebella.com /encyclopedia/article-Clique.html   (372 words)

 Encyclopedia: Clique A clique ('klik) is an informal and restricted social group formed by a number of people who share common interests - formal social groups are referred to as societies or organisations. In sociology, a group is usually defined as a collection consisting of a number of people who share certain aspects, interact with one another, accept rights and obligations as members of the group and share a common identity. A lawsuit is a civil action brought before a court in order to recover a right, obtain damages for an injury, obtain an injunction to prevent an injury, or obtain a declaratory judgment to prevent future legal disputes. www.nationmaster.com /encyclopedia/Clique   (1631 words)

 Talk:Clique problem - Wikipedia, the free encyclopedia I propose we move this to maximum clique problem, which is a commonly used name that is more accurate and specific. I removed the proof that maximum clique is NP-complete and replaced it by an appeal to the NP-completeness of independent set. Admittedly the proof on independent set problem isn't all that nice, and any improvement to it would be great. en.wikipedia.org /wiki/Talk:Clique_problem   (176 words)

 Lectures Week 1   (Site not responding. Last check: 2007-11-05) The CNF-Sat problem has as input a Boolean formula represented by a conjunction of clauses which are disjunctions, i.e., a CNF formula is of the form. A clique of size k in a graph G is a set C of k vertices in the graph such that every pair of vertices in C are connected by an edge in G, i.e., the subgraph induced by G on C is (isomorphic) to the the complete graph on k vertices. The clique problem: Input is a graph G with n vertices and a postive integer k, and the output answers the question whether G has a clique of size k. www.ececs.uc.edu /~annexste/Courses/cs782-2003/lec2.htm   (772 words)

 Volume 26: Cliques, Coloring and Satisfiability A maximum clique is, naturally, a clique whose number of vertices is at least as large as that for any other clique in the graph. While there are some relationships between the problems (clique size and coloring numbers provide bounds for each other, both clique and coloring problems can be formulated as satisfiability problem), most of the participants of the Challenge chose to concentrate on one of the problems. While clique algorithms were developed that could routinely solve graphs with hundreds of vertices, the graph coloring algorithms tended to be ineffective on random graphs with as few as seventy vertices. dimacs.rutgers.edu /Volumes/Vol26.html   (2676 words)

 Clique problem   (Site not responding. Last check: 2007-11-05) The clique problem is the optimization problem of finding a clique of maximum size in a graph (the maximal complete subgraph).The problem is a decision problem, so we wonder if a clique of size k exists in the graph. A brute force algorithm to find a clique in a graph is to list all subsetsof vertices, V and check each one to see if it forms a clique. A better one is to start with each node as a clique of one, and to merge cliques into larger cliques until there are no morepossible merges to check. www.therfcc.org /clique-problem-221615.html   (400 words)

 reductionhw   (Site not responding. Last check: 2007-11-05) The input to the clique problem is an undirected graph H and an integer j. The input to the three coloring problem is a graph G, and the problem is to decide whether the vertices of G can be colored with three colors such that no pair of adjacent vertices are colored the same color. The input to the four coloring problem is a graph G, and the problem is to decide whether the vertices of G can be colored with four colors such that no pair of adjacent vertices are colored the same color. www.dcc.uchile.cl /~cc30b/etc/guiaComplejidad/preg/index.html   (1644 words)

 [No title] The Set Splitting problem gives a set of elements and a collection of subsets, and asks if there is a way to split the set into two parts, such that each one of the subsets contains at least one element in each part. The Coloring Problem Prove that if the maximum degree of a graph is two or less, then you can solve the coloring problem in polynomial time. This can be written formally as a decision problem as follows: Input: A directed graph with positive and negative edge weights, two vertices x and y, and a positive integer bound M. Question: Is there a path from x to y such that the sum of the edge weights is between —M and M inclusive? aduni.org /courses/algorithms/courseware/psets/Problem_Set_06.doc   (588 words)

 Definition of problem Often, the causes of a problem are not known, in which case [[root cause analysi... The problem is a decision problem, so we wonder if a clique of size ''k'' exists in... 2: The '''Gettier problem''' is a fundamental problem in contemporary [[epistemology]] (the philosophy... www.wordiq.com /search/problem.html   (1003 words)

 [No title] Such a problem not necessarily in NP is called NP-Hard: the problem is neither solvable nor verifiable in poly-time. A computational problem (aka just "problem") Q can be viewed as a binary relation (on a set I of instances and a set S of solutions) that is a function although the function is not necessarily bijective or injective. Consider a corresponding verification problem (constrained version of HC): Given a solution path and a graph G, verify that path is a Hamiltonian cycle in G: verify path is a cycle and that it visits all vertices. ranger.uta.edu /~cook/aa/transcript/ln27f   (4598 words)

 [No title] Speaking of real life', although last week we restricted our discussions to decision problems as the theory of NP, P, and NP-completeness is stated in terms of decision problems, in actuality many of the decision problems we talked about show up as optimization problems'. For a minimization problem, we know that C(I)/C*(I) >= 1 for all I (as the optimal is the smallest); a ratio-bound r(n) means that for the worst input I of size n still the approximate solution C(I) is less than (or equal) to r(n) times the cost of the optimal solution C*(I). Indeed, optimization versions of NP-hard problems have widely different behavior with respect to approximation even though the decision versions all seem to be "as hard" as each other. theory.lcs.mit.edu /classes/6.046/spring04/lectures/lecture23.txt   (1094 words)

 [No title] These are problems for which the output is always either YES or NO. Formally, a decision problem D is a function from the set of all strings (all possible inputs) to the set {1,0}. Intuitively, we say that a decision problem D is in NP if (regardless whether it is easy or hard to solve) it has the following property: For every yes-input x to D, there is a polynomial size piece of evidence_x which can be checked in polynomial time that indeed x is a yes-input. Note that S=m as there is one vertex per clause in S. And its a clique because there is an edge between any pair of vertices in S as they all are consistent with the unique x1....xn and thus they are consistent between themselves. theory.lcs.mit.edu /classes/6.046/spring04/lectures/lecture20-21.txt   (1830 words)

 Citations: The maximum clique problem - Bomze, Budinich, Pardalos, Pelillo (ResearchIndex)   (Site not responding. Last check: 2007-11-05) The goal of the maximum clique problem is to nd a clique of maximum cardinality. From each node of the graph, a clique is incrementally built from candidates belonging to its adjacency list, starting with the node which has the largest intersection between its neighbors and the other candidates. A maximal clique is one which is not contained in any larger clique, while a maximum clique is a clique having the largest cardinality. citeseer.ist.psu.edu /context/45468/2030   (2023 words)

 Challenging Benchmarks for Maximum Clique, Maximum Independent Set, Minimum Vertex Cover and Vertex Coloring - ...   (Site not responding. Last check: 2007-11-05) 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. Given a graph G, a clique is a subset S of vertices in G such that each pair of vertices in S is connected by an edge. 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   (1624 words)

 Clique problem at opensource encyclopedia   (Site not responding. Last check: 2007-11-05) de:Cliquen und stabile Mengen In computer science, the Clique Problem is an NP-complete problem in complexity theory. The problem of finding $n$-element clique is equivalent to finding a set of literals satisfying SAT. And because there are no edges between literals of the same variable but different sign, if node of literal $x$ is in the clique, no node of literal of form $\neg x$ is. wiki.tatet.org /Clique_problem.html   (393 words)

 [No title] Haj\'os \cite3 later gave a combinatorial problem concerning factorization of abelian groups, which he proved was equivalent to Keller's conjecture. It has maximal clique size equal to the independent set number of $C_4\otimes \cdots \otimes C_4$, which is $\alpha(C_4)^n=2^n$ since $C_4$ is a perfect graph and $\alpha(C_4)=2$. In fact, $G_n$ has an enormous number of maximal cliques, and the problem is whether or not any of them remain a clique in $G^*_n$. www.ams.org /journals/bull/pre-1996-data/199227-2/Lagarias   (2370 words)

 A New Algorithm For The Maximum-Weight Clique Problem - Osterg (ResearchIndex)   (Site not responding. Last check: 2007-11-05) Abstract: Given a graph, in the maximum clique problem one wants to find the largest number of vertices, any two of which are adjacent. In the maximum-weight clique problem, the vertices have positive, integer weights, and one wants to find a clique with maximum weight. A recent algorithm for the maximum clique problem is here used as a basis for developing an algorithm for the weighted case. citeseer.ist.psu.edu /498302.html   (562 words)

 [No title] A semi-external memory {GRASP} is presented to approximately solve the maximum clique problem and maximum quasi-clique problem. In this paper, the facial structure of the polytope associated to the problem is studied and new classes of facies are introduced. The covering tour problem is an NP-hard problem, consisting in finding a minimum length tour or a minimum length Hamiltonian cycle over a subset of $V\cup W$ such that it includes all nodes in $T$ and covers all nodes in $W$. www.research.att.com /~mgcr/grasp/gannbib/bibtex/graphs.bib   (1811 words)

 reductionhwsolns   (Site not responding. Last check: 2007-11-05) To reduce the clique optimaization problem to the clique decision problem, we need to show that problem of finding a clique of maximum size is reducible to determine the size of the maximum clique. To reduce the Hamiltonian path problem to the k-degree MST problem, check to see if there is a MST with degree at most 2 (such a tree would be a Hamiltonian path). To reduce the 3-coloring problem to the 4-coloring problem, create a new vertex v that is connected to every other vertex in the original graph. www.dcc.uchile.cl /~cc30b/etc/guiaComplejidad/sols   (579 words)

 [No title]   (Site not responding. Last check: 2007-11-05) We measure the performance of a standard genetic algorithm on an elementary set of problem instances consisting of embedded cliques in random graphs. As we scale up the problem size and test on hard'' benchmark instances, we notice a degraded performance in the algorithm caused by premature convergence to local minima. To alleviate this problem, a sequence of modifications are implemented ranging from changes in input representation to systematic local search. www.cs.bu.edu /techreports/abstracts/1993-015   (215 words)

 Top: ClassW04ApproxAlgs/QiaofengYang This problem can be solved in polynomial time because maximum vertex biclique is equivalent to the maximum independent set in bipartite graphs which, in turn, can be solved via the minimum cut algorithm. Maximum Clique Number of graph G -- ω(G): the number of vertices in the maximum clique of G. Chromatic Number of graph G -- χ(G): the least number of colors for which the vertices of the graph G can be colored so that the two endpoints of any edge have different colors. The size of the Maximum Clique in graph G (Maximum Independent Set in the complement graph of G) is upper bounded by Lovasz theta function. www.cs.ucr.edu /~neal/wiki/wiki.pl?ClassW04ApproxAlgs/QiaofengYang   (1172 words)

 HW#3 Functional Decomposition Machine Design Using VHDL Maximum clique problem is basically a NP complete problem, there are many heuristics proposed. The Maximum clique problem can be also formulated as graph coloring problem, the graph for graph coloring problem will be complementary to the graph for the Maximum clique problem. We will use color to represent clique, i.e., the problem becomes find the maximum cliques in the graph and color vertices such that vertices in one clique have the same color. www.ee.pdx.edu /~mperkows/CLASS_VHDL/MA-99/max_cliq.html   (635 words)

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