Abstract
The Nemhauser&Trotter Theorem provides an algorithm which is frequently used as a subroutine in approximation algorithms for the classical Vertex Cover problem. In this paper we present an extension of this theorem so it fits a more general variant of Vertex Cover, namely the Generalized Vertex Cover problem, where edges are allowed not to be covered at a certain predetermined penalty. We show that many applications of the original Nemhauser&Trotter Theorem can be applied using our extension to Generalized Vertex Cover. These applications include a \((2-\frac{2}{d})\)-approximation algorithm for graphs of bounded degree d, a PTAS for planar graphs, a \((2-\frac{\lg \lg n}{2 \lg n})\)-approximation algorithm for general graphs, and a 2k kernel for the parameterized Generalized Vertex Cover problem.
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References
König, D.: Graphok és matrixok (Hungarian; Graphs and matrices). Matematikai és Fizikai Lapok 38, 116–119 (1931)
Karp, R.M.: Reducibility among combinatorial problems. Complexity of Computer Computations, 85–103 (1972)
Nemhauser, G.L., Trotter Jr., L.E.: Properties of vertex packing and independence system polyhedra. Math. Prog. 6, 48–61 (1974)
Hochbaum, D.S.: Approximation algorithms for the set covering and vertex cover problems. SIAM J. Comp. 11(3), 555–556 (1982)
Nemhauser, G.L., Trotter Jr., L.E.: Vertex packings: Structural properties and algorithms. Math. Prog. 8(2), 232–248 (1975)
Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 6, 243–254 (1983)
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–46 (1985)
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Applied Math. 36(2), 177–189 (1979)
Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)
Chen, J., Kanj, I.A., Jia, W.: Vertex cover: Further observations and further improvements. J. Alg. 41(2), 280–301 (2001)
Chlebík, M., Chlebíková, J.: Improvement of Nemhauser-Trotter Theorem and its applications in parametrized complexity. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 174–186. Springer, Heidelberg (2004)
Hochbaum, D.S.: Solving integer programs over monotone inequalities in three variables: a framework of half integrality and good approximations. European Journal of Operational Research 140(2), 291–321 (2002)
Bar-Yehuda, R., Rawitz, D.: On the equivalence between the primal-dual schema and the local ratio technique. SIAM J. Disc. Math. 19(3), 762–797 (2005)
Hassin, R., Levin, A.: The minimum generalized vertex cover problem. ACM Trans. Alg. 2(1), 66–78 (2006)
Hassin, R., Tamir, A.: Improved complexity bounds for location problems on the real line. Operations Research Letters 10, 395–402 (1991)
Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comp. 24(2), 296–317 (1995)
Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers. In: 12th SODA, pp. 642–651 (2001)
Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual-fitting with factor-revealing LP. J. ACM 50(6), 795–824 (2003)
Dinur, I., Safra, S.: The importance of being biased. In: 34th ACM Symposium on the Theory of Computing, pp. 33–42 (2002)
Könemann, J., Parekh, O., Segev, D.: A unified approach to approximating partial covering problems. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 468–479. Springer, Heidelberg (2006)
Bar-Yehuda, R.: One for the price of two: A unified approach for approximating covering problems. Algorithmica 27(2), 131–144 (2000)
Brooks, R.L.: On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosophical Society 37, 194–197 (1941)
Robertson, N., Seymour, P.D.: Graph minors. ii. algorithmic aspects of tree-width. J. Alg. 7(3), 309–322 (1986)
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Bar-Yehuda, R., Hermelin, D., Rawitz, D. (2010). Extension of the Nemhauser and Trotter Theorem to Generalized Vertex Cover with Applications. In: Bampis, E., Jansen, K. (eds) Approximation and Online Algorithms. WAOA 2009. Lecture Notes in Computer Science, vol 5893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12450-1_2
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DOI: https://doi.org/10.1007/978-3-642-12450-1_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-12449-5
Online ISBN: 978-3-642-12450-1
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