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A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set in Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4271))

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Abstract

A dominating set \(\mathcal{D}\) of a graph G=(V,E) is a subset of vertices such that every vertex in \(V \setminus \mathcal{D}\) has at least one neighbour in \(\mathcal{D}\). Moreover if \(\mathcal{D}\) is an independent set, i.e. no vertices in \(\mathcal{D}\) are pairwise adjacent, then \(\mathcal{D}\) is said to be an independent dominating set. Finding a minimum independent dominating set in a graph is an NP-hard problem. We give an algorithm computing a minimum independent dominating set of a graph on n vertices in time O(1.3575n). Furthermore, we show that Ω(1.3247n) is a lower bound on the worst-case running time of this algorithm.

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References

  1. Broersma, H., Kloks, T., Kratsch, D., Müller, H.: Independent sets in Asteroidal Triple-free graphs. SIAM J. Discrete Math. 12, 276–287 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chang, M.-S.: Efficient algorithms for the domination problems on interval and circular-arc graphs. SIAM J. Comput. 27, 1671–1694 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Corneil, D.-G., Perl, Y.: Clustering and domination in perfect graphs. Discrete Appl. Math. 9, 27–39 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  4. Farber, M.: Independent domination in chordal graphs. Operation Research Letters 1, 134–138 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and Conquer: A Simple O(20.288n) Independent Set Algorithm. In: Proceedings of SODA 2006, pp. 18–25 (2006)

    Google Scholar 

  6. Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: Domination - A case study. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 191–203. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  7. Fomin, F.V., Grandoni, F., Kratsch, D.: Some new techniques in design and analysis of exact (exponential) algorithms. Bulletin of the EATCS 87, 47–77 (2005)

    MATH  MathSciNet  Google Scholar 

  8. Fomin, F.V., Kratsch, D., Woeginger, G.J.: Exact (exponential) algorithms for the dominating set problem. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 245–256. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  9. Grandoni, F.: A note on the complexity of minimum dominating set. Discrete Algorithms 4, 209–214 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Halldórsson, M.M.: Approximating the Minimum Maximal Independence Number. Inf. Process. Lett. 46, 169–172 (1993)

    Article  MATH  Google Scholar 

  11. Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Inf. Process. Lett 27, 119–123 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kratsch, D., Stewart, L.: Domination on Cocomparability Graphs. SIAM J. Discrete Math. 6, 400–417 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. Moon, J.W., Moser, L.: On cliques in graphs. Israel J. Math. 3, 23–28 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  14. Randerath, B., Schiermeyer, I.: Exact algorithms for Minimum Dominating Set, Technical Report zaik-469, Zentrum fur Angewandte Informatik, Köln, Germany (April 2004)

    Google Scholar 

  15. Robson, J.M.: Algorithms for maximum independent sets. J. Algorithms 7, 425–440 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  16. Robson, J.M.: Finding a maximum independent set in time O(2n/4), Technical Report 1251-01, LaBRI, Université Bordeaux I (2001)

    Google Scholar 

  17. Tarjan, R.E., Trojanowski, A.E.: Finding a maximum independent set. SIAM J. Comput. 6, 537–546 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  18. Woeginger, G.J.: Exact algorithms for NP-hard problems: A survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, N-5020, Bergen, Norway

    Serge Gaspers

  2. Laboratoire d’Informatique Théorique et Appliquée, Université Paul, Verlaine – Metz, 57045 Cedex 01, Metz, France

    Mathieu Liedloff

Authors
  1. Serge Gaspers
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  2. Mathieu Liedloff
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Editors and Affiliations

  1. Department of Informatics, University of Bergen, N-5020, Bergen, Norway

    Fedor V. Fomin

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© 2006 Springer-Verlag Berlin Heidelberg

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Gaspers, S., Liedloff, M. (2006). A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set in Graphs. In: Fomin, F.V. (eds) Graph-Theoretic Concepts in Computer Science. WG 2006. Lecture Notes in Computer Science, vol 4271. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11917496_8

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  • DOI: https://doi.org/10.1007/11917496_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-48381-6

  • Online ISBN: 978-3-540-48382-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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