Journal of Combinatorial Optimization

, Volume 23, Issue 1, pp 94–117 | Cite as

The max quasi-independent set problem

  • N. Bourgeois
  • A. Giannakos
  • G. Lucarelli
  • I. Milis
  • V. T. Paschos
  • O. Pottié
Article

Abstract

In this paper, we deal with the problem of finding quasi-independent sets in graphs. This problem is formally defined in three versions, which are shown to be polynomially equivalent. The one that looks most general, namely, f-max quasi-independent set, consists of, given a graph and a non-decreasing function f, finding a maximum size subset Q of the vertices of the graph, such that the number of edges in the induced subgraph is less than or equal to f(|Q|). For this problem, we show an exact solution method that runs within time \(O^{*}(2^{\frac{d-27/23}{d+1}n})\) on graphs of average degree bounded by d. For the most specifically defined γ-max quasi-independent set and k-max quasi-independent set problems, several results on complexity and approximation are shown, and greedy algorithms are proposed, analyzed and tested.

Keywords

Quasi independent set Exact algorithms Approximation algorithms 

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References

  1. Abello J, Resende MGC, Sudarsky S (2002) Massive quasi-clique detection. In: Proceedings of LATIN 2002. LNCS, vol 2286. Springer, Berlin, pp 598–612 Google Scholar
  2. Asahiro Y, Iwama K, Tamaki H, Tokuyama T (2000) Greedily finding a dense subgraph. J Algorithms 34(2):203–221 CrossRefMATHMathSciNetGoogle Scholar
  3. Berge C (1973) Graphs and hypergraphs. North-Holland, Amsterdam MATHGoogle Scholar
  4. Boginski V, Butenko S, Pardalos P (2005) Mining market data: a network approach. Comput Oper Res. Available online at http://www.sciencedirect.com
  5. Corneil DG, Perl Y (1984) Clustering and domination in perfect graphs. Discrete Appl Math 9:27–39 CrossRefMATHMathSciNetGoogle Scholar
  6. Feige U, Kortsarz G, Peleg D (2001) The dense k-subgraph problem. Algorithmica 29(3):410–421 CrossRefMATHMathSciNetGoogle Scholar
  7. Fomin FV, Hoie K (2006) Pathwidth of cubic graphs and exact algorithms. Inf Process Lett 97(5):191–196 CrossRefMATHMathSciNetGoogle Scholar
  8. Goldberg AV (1984) Finding a maximum density subgraph. Technical report UCB CSD 84/171, University of California, Berkeley, CA Google Scholar
  9. Halldórsson MM, Radhakrishnan J (1994) Greed is good: approximating independent sets in sparse and bounded-degree graphs. In: Proceedings of STOC 1994, pp 439–448 Google Scholar
  10. Hartwell LH, Hopfield JJ, Leibler S, Murray AW (1999) From molecular to modular cell biology. Nature 402:C47–C52 CrossRefGoogle Scholar
  11. Hochbaum DS, Goldschmidt O (1997) k-edge subgraph problems. Discrete Appl Math 74(2):159–169 CrossRefMATHMathSciNetGoogle Scholar
  12. Jagota A, Narasimhan G, Šoltés Ľ (2001) A generalisation of maximal independent sets. Discrete Appl Math 109(3):223–235 CrossRefMATHMathSciNetGoogle Scholar
  13. Krishnamoorthy MS, Deo N (1979) Node-deletion NP-complete problems. SIAM J Comput 8:619–625 CrossRefMATHMathSciNetGoogle Scholar
  14. Picard JC, Queyranne M (1982) Selected applications of minimum cuts in networks. Inf Syst Oper Res 20:394–422 MATHGoogle Scholar
  15. Reed B (1996) Paths, stars and the number three. Comb Probab Comput 5:277–295 CrossRefMATHGoogle Scholar
  16. Yannakakis M, Lewis J (1980) The node-deletion problem for hereditary properties is NP-complete. J Comput Syst Sci 20(2):219–230 CrossRefMATHMathSciNetGoogle Scholar
  17. Zuckerman D (2006) Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proceedings of STOC 2006, pp 681–690 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • N. Bourgeois
    • 1
  • A. Giannakos
    • 1
  • G. Lucarelli
    • 1
    • 2
  • I. Milis
    • 2
  • V. T. Paschos
    • 1
  • O. Pottié
    • 1
  1. 1.LAMSADECNRS FRE 3234 and Université Paris-DauphineParisFrance
  2. 2.Department of InformaticsAthens University of Economics and BusinessAthensGreece

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