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Topic: Independent set problem

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Independent set

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	NationMaster - Encyclopedia: Independent Set problem
		In mathematics, the independent set problem (IS) is a well-known problem in graph theory and combinatorics.
		Given a graph G, an independent set is a subset of its vertices that are pairwise not adjacent.
		It fact, the independent set problem and the clique problem are equivalent, in the sense that if we know one is NP-complete, we can easily show that the other is NP-complete, and most algorithms for solving one problem can be transformed into an algorithm which solves the other in the same time and space.
	www.nationmaster.com /encyclopedia/Independent-Set-problem (1339 words)

	Independent Set
		Independent sets avoid conflicts between elements and hence arise often in coding theory and scheduling problems.
		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.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE173.HTM (635 words)

	Independent set - Wikipedia, the free encyclopedia
		The opposite of an independent set is a clique, in the sense that every independent set corresponds to a clique in the complement graph.
		It fact, the independent set problem and the clique problem are equivalent, in the sense that if we know one is NP-complete, we can easily show that the other is NP-complete, and most algorithms for solving one problem can be transformed into an algorithm which solves the other in the same time and space.
		The problem of finding a maximal independent set can be solved in polynomial time by a trivial greedy algorithm.
	en.wikipedia.org /wiki/Independent_set (274 words)

	Independent Set (Site not responding. Last check: 2007-10-20)
		Independent sets avoid conflicts between elements and hence arise often in coding theory and scheduling problems.
		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)

	CS523: Growing practice problem set 2 (Site not responding. Last check: 2007-10-20)
		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.
		Independent set: An independent set of a graph G = (V, E) is a subset V' of V such that each edge in E is incident on at most one vertex in V'.
		Independent set problem: Given a graph G and a positive integet k, the independent set problem asks whether an independent set of size k exists in G. Show that the independent set problem is NP-hard using a reduction from:
	web.engr.oregonstate.edu /~saurabh/cs523/practice3.html (701 words)

	CSCI 3434 - NP Problems (Site not responding. Last check: 2007-10-20)
		If the set is independent, then the algorithm answers "Yes"; otherwise, the algorithm answers "No." If at least one of the nondeterministic guesses is an independent set, then the result of the nondeterministic algorithm will be "Yes".
		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.
	www.cs.colorado.edu /~main/theory/s3-np.html (1263 words)

	Challenging Benchmarks for Maximum Clique, Maximum Independent Set, Minimum Vertex Cover and Vertex Coloring - ...
		A vertex cover (or hitting set) 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 (1663 words)

	Citations: the complexity of approximating the independent set problem - Berman, Schnitger (ResearchIndex)
		On the complexity of approximating the independent set problem.
		Reduce the problem of approximating max 3SAT to approximating ff(G 0) within a constant factor in a graph G 0 that has linear size independent sets.
		The problem of approximating ff(G 0) within a constant factor is reduced to that of approximating ff(G) within a factor of O(N ffl) for some ffl 0.
	citeseer.ist.psu.edu /context/30338/0 (2288 words)

	CSCI 3650 Answers to practice questions for exam 3
		By the way: If A is a set of strings then the complement of a set A is the set of all strings that do not belong to A. The same kind argument does not work to show that if A is in NP then the complement of A is also in NP.
		An independent set in a graph is a set of vertices that are mutually nonadjacent.
		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)

	Anatoly Plotnikov's selected papers
		The main result is that this problem can be considered as the solving the same problem in a subclass of the weighted normal twin-orthogonal graphs.
		We introduce two auxiliary sets that characterize the solution of the problem: the adjoint set, which contains the elements from the original set none of which can be adjoined to the already chosen solution elements; and the residual set, in which every element can be adjoined to previously chosen solution elements.
		It is proven that a graph is Hamiltonian if and only if the number of elements in any its independent set of the vertices does not exceed the number of vertices in the minimum separating set for paths, which connect this independent vertices between themselves.
	www.vinnica.ua /~aplot/papers.html (583 words)

	Citations: A fast and simple randomized parallel algorithm for the maximal independent set problem - Alon, Babai, Itai ...
		However, it was soon realized that O(1) wise independent binary random variable constructions were inadequate to derandomize many parallel algorithms; n random variables that are roughly O(log n) wise independent seemed to....
		S is a maximal independent set (MIS) if no proper superset of S is an IS. It is easy to find an MIS sequentially, but efficient parallel algorithms appear much harder.
		It is easy to see that this class maps each d elements of the domain independently to the range, and thus, the bound that applies to the class of all functions also applies to this class.
	citeseer.ist.psu.edu /context/30310/0 (1698 words)

	Approach:
		An independent set is defined as a subset that is joined by no more than two vertices in an edge.
		Our results showed that the Maximum Independent Set of a graph is related to the degree of density in the graph.
		This problem shows why parallel computing is useful when it comes to exhaustive searching problems like the Maximum Independent Set problem, Traveling Salesman Problem, and Graph Coloring Problem.
	www.people.virginia.edu /~nem4e/cse457_final/457_final.html (1313 words)

	COSC 4111.03 -- Assignment 3 (Site not responding. Last check: 2007-10-20)
		Prove the INDEPENDENT SET problem is NP-complete by a reduction from 3SAT.
		A dominating set in an undirected graph G is a subset of vertices such that every vertex is either in the subset, or connected to a vertex in the subset by an edge.
		In the n point traveling salesman problem, we are given an n x n cost matrix C[i,j] of nonnegative integers, specifying the cost of traveling directly between points i and j.
	www.cs.yorku.ca /course_archive/2003-04/F/4111/2003.a3.html (363 words)

	ISMP 2000 - Meeting Topics (Site not responding. Last check: 2007-10-20)
		A combinatorial approach to the DNAsequencing problem (the problem of identifying similarities among genes) is described.
		In this work, we consider the natural generalization of the problem arising when one is given many signed permutations (genomes) and would like to find the signed permutation at minimum reversal distance from the given ones, i.e., the permutation whose sum of distances from all the given permutations is a minimum.
		We propose a natural graph-theoretic relaxation of this problem, calling for a matching with a special structure in a graph which forms the maximum number of cycles with a set of given matchings (corresponding to the given permutations).
	www.isye.gatech.edu /ismp2000/schedule/session_pages/THC-20-SC319.html (427 words)

	Abstracts
		The lattice generated by a set S of rational vectors is the set of linear integer combinations of those vectors.
		One fundamental computational problem concerning lattices is the closest lattice vector problem: given the basis of a lattice, and given a vector x not in the lattice, find the vector in the lattice that is closest to x.
		We give an algorithm that solves the problem; however, the time required by the algorithm depends on the distance between x and the lattice, and on the quality of the basis.
	www.cis.temple.edu /~beigel/FESNP/abstracts.html (1529 words)

	Independent set problem - Definition, explanation
		In mathematics, the independent set problem (\IS) is a question in graph theory and combinatorics, known to be an NP-complete problem.
		Then, the independent set problem asks if, given a graph G and an integer k, does G have an independent set of size at least k?
		This problem is known to have no constant-factor approximation algorithm unless P=NP.
	www.calsky.com /lexikon/en/txt/i/in/independent_set_problem.php (773 words)

	SAT01 Notes
		Sometimes (1) is introduced to be a special satisfiability problem, whose literals do not have negations and where such an assignment of the variables has to be found that each clause has exactly one true literal.
		In general, we consider any problem to be a certain set of contradictions among demands that exist in the universe of the problem.
		A problem (in the broad sense, including non-mathematical) is an equation P(x) = true where x is a looked-for object of some universe and P is a given predicate defined over the universe.
	www.geocities.com /st_busygin/sat01_notes.html (4379 words)

	Stas Busygin's NP-Completeness Page (Site not responding. Last check: 2007-10-20)
		A problem of a class is complete if you can solve any other problem of this class in polynomial time having a polynomial time algorithm for the first one.
		Hence complete problems are hardest in their own classes and as they exist we may choose any of them to advance solving techniques for the entire class.
		The concept of complete problems for a class is generalized to hard problems for the class by inclusion of all other problems, whose polynomial time algorithm gives polynomial time solvability for the class.
	www.busygin.dp.ua /npc.html (896 words)

	[No title]
		As an example of the input file needed by the code, as well as the output generated by it, consider the maximum independent set instance whose input file (sample.dat) is in the distribution.
		An independent set of size 7 was found in the first GRASP iteration.
		c Either it is in the independent set or is adjacent to a c vertex that is in the independent set.
	www.netlib.org /toms/787 (4051 words)

	Mathematical Programming Glossary - I (Site not responding. Last check: 2007-10-20)
		Given a graph, G=[V,E], an independent set (also called a stable set) is a subgraph induced by a subset of vertices, S, plus all edges with both endpoints in S, such that no two nodes in the subgraph are adjacent.
		A maximal independent set is one such that adding any node causes the subgraph to violate independence -- the added node is adjacent to some node already in the set.
		Given weights, {w(v)} for v in V, the weight of an independent set, S, is the sum of the weights in S. The maximum independent set problem is to find an independent set of maximum weight.
	carbon.cudenver.edu /~hgreenbe/glossary/I.html (1523 words)

	Independent Set and Vertex Cover (Site not responding. Last check: 2007-10-20)
		The maximum independent set decision problem is formally defined:
		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.
	www.cs.toronto.edu /~yuana/AlgorithmManual/BOOK/BOOK3/NODE108.HTM (320 words)

	CodeGuru Forums - A little help on an NP-Complete Problem (Site not responding. Last check: 2007-10-20)
		The Independent-Set is to find a maximum-size independent set in G. a) Formulate a related decision problem for the independent-set problem, and prove that it is NP-complete.
		Although the independent-set decision problem is NP-complete, certain special cases are polynominal-time solvable.
		The problem is that among these people there are some that are enemies.
	www.codeguru.com /forum/archive/index.php/t-301401.html (425 words)

	Project
		Given an undirected graph G=(V,E), an independent set is a set I of vertices such that for every pair of vertices i,j in I, the edge (i,j) does not belong to E. The maximum indpendent set problem is the problem of finding a largest independent set in a given graph.
		This problem is very hard in general, however it can be solved in O(V) time on acyclic graphs (trees and forests).
		Your program outputs the size of the largest independent set in the graph (or 0 if the graph has a cycle) on the standard output.
	www.cs.berkeley.edu /~luca/cs170/project (653 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)

	Teaching Textbooks
		Each of the 5 practice problems is labeled with a letter (a, b, c, d, or e).
		These problems are very similar to the problems in the problem set that are labeled with those same letters.
		In fact, nearly every problem set includes several problems that were modeled after those found on actual SAT and ACT exams.
	www.teachingtextbooks.com /FAQs.htm (3032 words)

	Assignment 13 for CSCI 256 (Site not responding. Last check: 2007-10-20)
		Show by reduction from CLIQUE that the INDEPENDENT SET problem is NP-complete.
		INDEPENDENT SET: An independent set in a graph G = (V,E) is a set of vertices, no two of which are connectied.
		The problem is to determine, given G and an integer k, whether G contains an independent set with >= k vertices.
	www.cs.williams.edu /~kim/cs256/CS256Assn13.html (167 words)

	1.5.2 Independent Set (Site not responding. Last check: 2007-10-20)
		Problem: What is the largest subset of vertices of V such that no pair of vertices defines an edge of E?
		Excerpt from The Algorithm Design Manual: The need to find large independent sets typically arises in dispersion problems, where we seek a set of mutually separated points.
		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.
	www.cs.sunysb.edu /~algorith/files/independent-set.shtml (246 words)

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