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Critical edges/nodes for the minimum spanning tree problem: complexity and approximation

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Abstract

In this paper, we study the complexity and the approximation of the k most vital edges (nodes) and min edge (node) blocker versions for the minimum spanning tree problem (MST). We show that the k most vital edges MST problem is NP-hard even for complete graphs with weights 0 or 1 and 3-approximable for graphs with weights 0 or 1. We also prove that the k most vital nodes MST problem is not approximable within a factor n 1−ϵ, for any ϵ>0, unless NP=ZPP, even for complete graphs of order n with weights 0 or 1. Furthermore, we show that the min edge blocker MST problem is NP-hard even for complete graphs with weights 0 or 1 and that the min node blocker MST problem is NP-hard to approximate within a factor 1.36 even for graphs with weights 0 or 1.

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References

  • Arora S, Lund C (1996) Hardness of approximations. In: Approximation algorithms for NP-hard problems. PWS Publishing Company, Boston, pp 399–446

    Google Scholar 

  • Bar-Noy A, Khuller S, Schieber B (1995) The complexity of finding most vital arcs and nodes. Technical report CS-TR-3539, University of Maryland

  • Bazgan C, Toubaline S, Vanderpooten D (2010) Complexity of determining the most vital elements for the 1-median and 1-center location problems. In: Proceeding of the 4th annual international conference on combinatorial optimization and applications (COCOA 2010), Part I. LNCS, vol 6508, pp 237–251

    Chapter  Google Scholar 

  • Bazgan C, Toubaline S, Vanderpooten D (2011) Efficient algorithms for finding the k most vital edges for the minimum spanning tree problem. In: Proceeding of the 5th annual international conference on combinatorial optimization and applications (COCOA 2011). LNCS, vol 6831, pp 126–140

    Chapter  Google Scholar 

  • Dinur I, Safra S (2005) On the hardness of approximating minimum vertex cover. Ann Math 162(1):439–485

    Article  MathSciNet  MATH  Google Scholar 

  • Frederickson GN, Solis-Oba R (1996) Increasing the weight of minimum spanning trees. In: Proceedings of the 7th ACM-SIAM symposium on discrete algorithms (SODA 1996), pp 539–546. Also appeared in J Algorithms 33(2):244–266 (1999)

    Google Scholar 

  • Håstad J (1999) Clique is hard to approximate within n 1−ε. Acta Math 182(1):105–142

    Article  MathSciNet  MATH  Google Scholar 

  • Hsu L, Jan R, Lee Y, Hung C, Chern M (1991) Finding the most vital edge with respect to minimum spanning tree in a weighted graph. Inf Process Lett 39(5):277–281

    Article  MathSciNet  MATH  Google Scholar 

  • Iwano K, Katoh N (1993) Efficient algorithms for finding the most vital edge of a minimum spanning tree. Inf Process Lett 48(5):211–213

    Article  MathSciNet  MATH  Google Scholar 

  • Khachiyan L, Boros E, Borys K, Elbassioni K, Gurvich V, Rudolf G, Zhao J (2008) On short paths interdiction problems: total and node-wise limited interdiction. Theory Comput Syst 43(2):204–233

    Article  MathSciNet  Google Scholar 

  • Khanna S, Motwani R, Sudan M, Vazirani U (1994) On syntactic versus computational views of approximability. In: Proceedings of the 35th annual IEEE annual symposium on foundations of computer science (FOCS 1994), pp 819–830. Also published in SIAM J Comput 28(1):164–191 (1999)

    Chapter  Google Scholar 

  • Liang W (2001) Finding the k most vital edges with respect to minimum spanning trees for fixed k. Discrete Appl Math 113(2–3):319–327

    Article  MathSciNet  MATH  Google Scholar 

  • Liang W, Shen X (1997) Finding the k most vital edges in the minimum spanning tree problem. Parallel Comput 23(3):1889–1907

    Article  MathSciNet  MATH  Google Scholar 

  • Nardelli E, Proietti G, Widmayer P (2001) A faster computation of the most vital edge of a shortest path. Inf Process Lett 79(2):81–85

    Article  MathSciNet  MATH  Google Scholar 

  • Ratliff HD, Sicilia GT, Lubore SH (1975) Finding the n most vital links in flow networks. Manag Sci 21(5):531–539

    Article  MathSciNet  MATH  Google Scholar 

  • Shen H (1999) Finding the k most vital edges with respect to minimum spanning tree. Acta Inform 36(5):405–424

    Article  MathSciNet  MATH  Google Scholar 

  • Suraweera F, Maheshwari P, Bhattacharya P (1995) Optimal algorithms to find the most vital edge of a minimum spanning tree. Technical report CIT-95-21, School of Computing and Information Technology, Griffith University

  • Wollmer R (1964) Removing arcs from a network. Oper Res 12(6):934–940

    Article  MathSciNet  MATH  Google Scholar 

  • Wood RK (1993) Deterministic network interdiction. Math Comput Model 17(2):1–18

    Article  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. LAMSADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Cristina Bazgan, Sonia Toubaline & Daniel Vanderpooten

  2. Institut Universitaire de France, Paris, France

    Cristina Bazgan

Authors
  1. Cristina Bazgan
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  2. Sonia Toubaline
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  3. Daniel Vanderpooten
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Correspondence to Cristina Bazgan.

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Bazgan, C., Toubaline, S. & Vanderpooten, D. Critical edges/nodes for the minimum spanning tree problem: complexity and approximation. J Comb Optim 26, 178–189 (2013). https://doi.org/10.1007/s10878-011-9449-4

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  • DOI: https://doi.org/10.1007/s10878-011-9449-4

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