Advertisement

Complexity of Most Vital Nodes for Independent Set in Graphs Related to Tree Structures

  • Cristina Bazgan
  • Sonia Toubaline
  • Zsolt Tuza
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)

Abstract

Given an undirected graph with weights on its vertices, the k most vital nodes independent set problem consists of determining a set of k vertices whose removal results in the greatest decrease in the maximum weight of independent sets. We also consider the complementary problem, minimum node blocker independent set that consists of removing a subset of vertices of minimum size such that the maximum weight of independent sets in the remaining graph is at most a specified value. We show that these problems are NP-hard on bipartite graphs but polynomial-time solvable on unweighted bipartite graphs. Furthermore, these problems are polynomial also on graphs of bounded treewidth and cographs. A result on the non-existence of a ptas is presented, too.

Keywords

most vital nodes independent set complexity NP-hard ptas bipartite graph bounded treewidth cograph 

Mathematics Subject Classification

05C85 05C69 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bar-Noy, A., Khuller, S., Schieber, B.: The complexity of finding most vital arcs and nodes, Technical Report CS-TR-3539, Department of Computer Science, University of Maryland (1995)Google Scholar
  2. 2.
    Bazgan, C., Toubaline, S., VanderPoten, D.: Détermination des éléments les plus vitaux pour le problème d’affectation. In: Actes de la 10ème Conférence de la Société Française de Recherche Opérationelle et d’Aide à la Décision, ROADEF 2010 (2010)Google Scholar
  3. 3.
    Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)CrossRefzbMATHGoogle Scholar
  4. 4.
    Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM Journal on Computing 14(4), 926–934 (1985)CrossRefzbMATHGoogle Scholar
  5. 5.
    Frederickson, G.N., Solis-Oba, R.: Increasing the weight of minimum spanning trees. In: Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1996), pp. 539–546 (1996)Google Scholar
  6. 6.
    Habib, M., Paul, C.: A simple linear time algorithm for cograph recognition. Discrete Applied Mathematics 145(2), 183–197 (2005)CrossRefzbMATHGoogle Scholar
  7. 7.
    Hassin, R.: Approximation schemes for the restricted shortest path. Mathematics of Operations Research 17(1), 36–42 (1992)CrossRefzbMATHGoogle Scholar
  8. 8.
    Khachiyan, L., Boros, E., Borys, K., Elbassioni, K., Gurvich, V., Rudolf, G., Zhao, J.: On short paths interdiction problems: total and node-wise limited interdiction. Theory of Computing Systems 43(2), 204–233 (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Kloks, T.: Treewidth, computations and approximations. LNCS, vol. 842. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  10. 10.
    Kőnig, D.: Graphs and matrices. Math. Fiz. Lapok 38, 116–119 (1931) (in Hungarian)zbMATHGoogle Scholar
  11. 11.
    Nemhauser, G.L., Trotter, L.E.: Vertex packing: structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)CrossRefzbMATHGoogle Scholar
  12. 12.
    Papadimitriou, C., Yannakakis, M.: Optimization, approximation and complexity classes. Journal of Computer and System Science 43(3), 425–440 (1991)CrossRefzbMATHGoogle Scholar
  13. 13.
    Petrank, E.: The hardness of approximation: gap location. Computational Complexity 4, 133–157 (1994)CrossRefzbMATHGoogle Scholar
  14. 14.
    Ries, B., Bentz, C., Picouleau, C., de Werra, D., Costa, M., Zenklusen, R.: Blockers and Transversals in some subclasses of bipartite graphs: When caterpillars are dancing on a grid. Discrete Mathematics 310(1), 132–146 (2010)CrossRefzbMATHGoogle Scholar
  15. 15.
    Shen, H.: Finding the k most vital edges with respect to minimum spanning tree. Acta Informatica 36, 405–424 (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Zenklusen, R., Ries, B., Picouleau, C., de Werra, D., Costa, M., Bentz, C.: Blockers and Transversals. Discrete Mathematics 309(13), 4306–4314 (2009)CrossRefzbMATHGoogle Scholar
  17. 17.
    Wood, R.K.: Deterministic network interdiction. Mathematical and Computer Modeling 17(2), 1–18 (1993)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Cristina Bazgan
    • 1
  • Sonia Toubaline
    • 1
  • Zsolt Tuza
    • 2
    • 3
  1. 1.LAMSADEUniversité Paris-DauphineFrance
  2. 2.Computer and Automation InstituteHungarian Academy of SciencesBudapestHungary
  3. 3.Department of Computer Science and Systems TechnologyUniversity of VeszprémHungary

Personalised recommendations