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Vertex Cover in Conflict Graphs: Complexity and a Near Optimal Approximation

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Combinatorial Optimization and Applications

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9486))

Abstract

Given finite number of forests of complete multipartite graph, conflict graph is a sum graph of them. Graph of this class can model many natural problems, such as in database application and others. We show that this property is non-trivial if limiting the number of forests of complete multipartite graph, then study the problem of vertex cover on conflict graph in this paper. The complexity results list as follow,

  • If the number of forests of complete multipartite graph is fixed, conflict graph is non-trivial property, but finding 1.36-approximation algorithms is NP-hard.

  • Given 2 forests of complete multipartite graph and maximum degree less than 7, vertex cover problem of conflict graph is NP-complete. Without the degree restriction, it is shown to be NP-hard to find an algorithm for vertex cover of conflict graph within \(\frac{17}{16}-\varepsilon \), for any \(\varepsilon >0\).

Given conflict graph consists of r forests of complete multipartite graph, we design an approximation algorithm and show that the approximation ratio can be bounded by \(2-\frac{1}{2^r}\). Furthermore, under the assumption of UGC, the approximation algorithm is shown to be near optimal by proving that, it is hard to improve the ratio with a factor independent of the size (number of vertex) of conflict graph.

This work was supported in part by the National Grand Fundamental Research 973 Program of China under grant 2012CB316200, the Major Program of National Natural Science Foundation of China under grant 61190115, Project 61502121 supported by National Natural Science Foundation of China.

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Notes

  1. 1.

    A more strict definition is based on the assumption that \(\varphi \) is a conjunction of relation atoms, and the assumption will not affect the expressability of dependencies.

References

  1. Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Boston (1995)

    MATH  Google Scholar 

  2. Ásgeirsson, E.I., Stein, C.: Divide-and-conquer approximation algorithm for vertex cover. SIAM J. Discret. Math. 23(3), 1261–1280 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Ann. Discrete Math. 25, 27–46 (1985)

    MathSciNet  MATH  Google Scholar 

  4. Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162, 439–485 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Comput. 1(2), 180–187 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. Håstad, J.: Some optimal inapproximability results. J. ACM (JACM) 48(4), 798–859 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Karakostas, G.: A better approximation ratio for the vertex cover problem. ACM Trans. Algorithms 5(4), 41:1–41:8 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2 - \(\epsilon \). J. Comput. Syst. Sci. 74(3), 335–349 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kuhn, F., Mastrolilli, M.: Vertex cover in graphs with locally few colors. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 498–509. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  10. Monien, B., Speckenmeyer, E.: Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Inform. 22(1), 115–123 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nemhauser, G.L., Trotter Jr, L.E.: Vertex packings: structural properties and algorithms. Math. Program. 8(1), 232–248 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sitton, D.: Maximum matchings in complete multipartite graphs. Furman Univ. Electron. J. Undergraduate Math. 2, 6–16 (1996)

    Google Scholar 

  13. Williamson, D.P., Shmoys, D.B.: The design of approximation algorithms. Cambridge University Press, New York (2011)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Harbin Institute of Technology, Harbin, 150001, Heilongjiang, China

    Dongjing Miao, Jianzhong Li, Xianmin Liu & Hong Gao

Authors
  1. Dongjing Miao
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  2. Jianzhong Li
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  3. Xianmin Liu
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  4. Hong Gao
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Corresponding author

Correspondence to Dongjing Miao .

Editor information

Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Marywood University, Scranton, Pennsylvania, USA

    Zaixin Lu

  2. North Carolina Central University, Durham, North Carolina, USA

    Donghyun Kim

  3. University of Texas at Dallas, Richardson, Texas, USA

    Weili Wu

  4. Texas Southern University, Houston, Texas, USA

    Wei Li

  5. University of Texas at Dallas, Richardson, Texas, USA

    Ding-Zhu Du

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Miao, D., Li, J., Liu, X., Gao, H. (2015). Vertex Cover in Conflict Graphs: Complexity and a Near Optimal Approximation. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_29

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  • DOI: https://doi.org/10.1007/978-3-319-26626-8_29

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26625-1

  • Online ISBN: 978-3-319-26626-8

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