A primal-dual approach to approximation of node-deletion problems for matroidal properties

Extended abstract
  • Toshihiro Fujito
Session 19: Algorithms IV
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1256)


This paper is concerned with the polynomial time approximability of node-deletion problems for hereditary properties. We will focus on such graph properties that are derived from matroids definable on the edge set of any graph. It will be shown first that all the node-deletion problem for such properties can be uniformly formulated by a simple but non-standard form of the integer program. A primaldual approximation algorithm based on this and the dual of its linear relaxation is then presented.

When a property has infinitely many minimal forbidden graphs no constant factor approximation for the corresponding node-deletion problem has been known except for the case of the Feedback Vertex Set (FVS) problem in undirected graphs. It will be shown next that FVS is not the sole exceptional case and that there exist infinitely many graph (hereditary) properties with an infinite number of minimal forbidden graphs, for which the node-deletion problems are efficiently approximable to within a factor of 2. Such properties are derived from the notion of matroidal families of graphs and relaxing the definitions for them.


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  1. [ALM+02]
    S. Arora, C. Lund. R. Motwani. M. Sudan, and M. Szegedy. Proof verification and hardness of approximation problems. In 33rd FOCS, pages 14–23. 1992.Google Scholar
  2. [And78]
    T. Andreae. Matroidal families of finite connected nonhomeomorphic graphs exist. J. of Graph Theory. 2:149–153, 1978.Google Scholar
  3. [BBF95]
    V. Bafna, P. Berman, and T. Fujito. Constant ratio approximations of the weighted feedback vertex set problem for undirected graphs. In ISAAC '95. pages 142–151, 1995.Google Scholar
  4. [BE85]
    R. Bar-Yehuda and S. Even. A local-ratio theorem for approximating the weighted vertex cover problem. In Annals of Discrete Mathematics, volume 25, pages 27–46. North-Holland, 1985.Google Scholar
  5. [BG94]
    A. Becker and D. Geiger. Approximation algorithms for the loop cutset problem. In Proc. of the 10th conference on Uncertainty in Artificial Intelligence, pages 60–68, 1994.Google Scholar
  6. [BGNR94]
    R. Bar-Yehuda, D. Geiger, J. Naor, and R. M. Roth. Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and bayesian inference. In 5th SODA, pages 344–354. 1994.Google Scholar
  7. [CGHW96]
    F.A. Chudak, M.X. Goemans, D.S. Hochbaum, and D.P. Williamson. A primal-dual interpretation of recent 2-approximation algorithms for the feedback vertex set problem in undirected graphs. Manuscript. 1996.Google Scholar
  8. [ENSZ96]
    G. Even, J. Naor, B. Schiever, and L. Zosin. Approximating minimum subset feedback sets in undirected graphs with applications. In 4th ISTCS, pages 78–88, 1996.Google Scholar
  9. [Fuj96]
    T. Fujito. A unified local ratio approximation of node-deletion problems. In ESA '96, pages 167–178, 1996.Google Scholar
  10. [Gav74]
    F. Gavril, 1974. cited in [GJ79, page 134].Google Scholar
  11. [G.J79]
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and co., New York. 1979.Google Scholar
  12. [Hås97]
    J. Håstad. Some optimal in-approximability results. In 29th STOC, to appear, 1997.Google Scholar
  13. [LY80]
    J.M. Lewis and M. Yannakakis. The node-deletion problem for hereditary properties is NP-complete. JCSS, 20:219–230, 1980.Google Scholar
  14. [LY93]
    C. Lund and M. Yannakakis. The approximation of maximum subgraph problems. In 20th ICALP, pages 40–51, 1993.Google Scholar
  15. [MS85]
    B. Monien and E. Speckenmeyer. Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Inform., 22:115–123, 1985.CrossRefGoogle Scholar
  16. [PY91]
    C. Papadimitriou and M. Yannakakis. Optimization, approximation and complexity classes. JCSS, 43:425–440, 1991.Google Scholar
  17. [Sim72]
    J.M.S. Simões-Pereira. On subgraphs as matroid cells. Math. Z., 127:315–322, 1972.CrossRefGoogle Scholar
  18. [Sim73]
    J.M.S. Simões-Pereira. On matroids on edge sets of graphs with connected subgraphs as circuits. In Proc. Amer. Math. Soc., volume 38, pages 503–506. 1973.Google Scholar
  19. [Wel76]
    D.J.A. Welsh. Matroid Theory. Academic Press, London, 1976.Google Scholar
  20. [Yan94]
    M. Yannakakis. Some open problems in approximation. In CIAC '94, pages 33–39, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Toshihiro Fujito
    • 1
  1. 1.Dept. of Electrical Engineering, Faculty of EngineeringHiroshima UniversityHigashi-HiroshimaJapan

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