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Vertex cover

About: Vertex cover is a(n) research topic. Over the lifetime, 3458 publication(s) have been published within this topic receiving 91497 citation(s). more


Open accessJournal ArticleDOI: 10.1016/S0022-0000(74)80044-9
David S. Johnson1Institutions (1)
Abstract: Simple, polynomial-time, heuristic algorithms for finding approximate solutions to various polynomial complete optimization problems are analyzed with respect to their worst case behavior, measured by the ratio of the worst solution value that can be chosen by the algorithm to the optimal value. For certain problems, such as a simple form of the kanpsack problem and an optimization problem based on satisfiability testing, there are algorithms for which this ratio is bounded by a constant, independent of the problem size. For a number of set covering problems, simple algorithms yield worst case ratios which can grow with the log of the problem size. And for the problem of finding the maximum clique in a graph, no algorithm has been found for which the ratio does not grow at least as fast as n^@e, where n is the problem size and @e>0 depends on the algorithm. more

Topics: Approximation algorithm (67%), Optimization problem (65%), Vertex cover (62%) more

2,411 Citations

Open accessJournal ArticleDOI: 10.1016/0304-3975(76)90059-1
Abstract: It is widely believed that showing a problem to be NP -complete is tantamount to proving its computational intractability. In this paper we show that a number of NP -complete problems remain NP -complete even when their domains are substantially restricted. First we show the completeness of Simple Max Cut (Max Cut with edge weights restricted to value 1), and, as a corollary, the completeness of the Optimal Linear Arrangement problem. We then show that even if the domains of the Node Cover and Directed Hamiltonian Path problems are restricted to planar graphs, the two problems remain NP -complete, and that these and other graph problems remain NP -complete even when their domains are restricted to graphs with low node degrees. For Graph 3-Colorability, Node Cover, and Undirected Hamiltonian Circuit, we determine essentially the lowest possible upper bounds on node degree for which the problems remain NP -complete. more

Topics: Maximum cut (60%), Forbidden graph characterization (59%), Hamiltonian path problem (58%) more

2,062 Citations

Journal ArticleDOI: 10.1145/502090.502098
Johan Håstad1Institutions (1)
01 Jul 2001-Journal of the ACM
Abstract: We prove optimal, up to an arbitrary e > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover. more

Topics: Vertex cover (55%), Approximation algorithm (54%), Unique games conjecture (53%) more

1,813 Citations

Open accessBook
01 Jan 2006-
Abstract: PART I: FOUNDATIONS 1. Introduction to Fixed-Parameter Algorithms 2. Preliminaries and Agreements 3. Parameterized Complexity Theory - A Primer 4. Vertex Cover - An Illustrative Example 5. The Art of Problem Parameterization 6. Summary and Concluding Remarks PART II: ALGORITHMIC METHODS 7. Data Reduction and Problem Kernels 8. Depth-Bounded Search Trees 9. Dynamic Programming 10. Tree Decompositions of Graphs 11. Further Advanced Techniques 12. Summary and Concluding Remarks PART III: SOME THEORY, SOME CASE STUDIES 13. Parameterized Complexity Theory 14. Connections to Approximation Algorithms 15. Selected Case Studies 16. Zukunftsmusik References Index more

Topics: Vertex cover (60%), Parameterized complexity (60%), Approximation algorithm (57%) more

1,686 Citations

Journal ArticleDOI: 10.1145/278298.278306
Sanjeev Arora1, Carsten Lund2, Rajeev Motwani3, Madhu Sudan4  +1 moreInstitutions (4)
01 May 1998-Journal of the ACM
Abstract: We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof” with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [1998] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length).As a consequence, we prove that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP = P. The class MAX SNP was defined by Papadimitriou and Yannakakis [1991] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige et al. [1996] and Arora and Safra [1998] and show that there exists a positive e such that approximating the maximum clique size in an N-vertex graph to within a factor of Ne is NP-hard. more

Topics: Probabilistically checkable proof (65%), Interactive proof system (61%), PCP theorem (58%) more

1,395 Citations

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Topic's top 5 most impactful authors

Saket Saurabh

93 papers, 3.5K citations

Michael R. Fellows

59 papers, 2.6K citations

Rolf Niedermeier

49 papers, 3.6K citations

Venkatesh Raman

44 papers, 1.4K citations

Daniel Lokshtanov

42 papers, 2K citations

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