A polyhedral approach to the feedback vertex set problem

  • Meinrad Funke
  • Gerhard Reinelt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1084)

Abstract

Feedback problems consist of removing a minimal number of arcs or nodes of a directed or undirected graph in order to make it acyclic. In this paper we consider a special variant, namely the problem of finding a maximum weight node induced acyclic subdigraph. We discuss valid and facet defining inequalities for the associated polytope and present computational results with a branch-and-cut algorithm.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cplex. Using the Cplex callable library and Cplex mixed integer library (1994). Cplex Inc.Google Scholar
  2. 2.
    Grötschel, M., Jünger, M., Reinelt, G.: On the acyclic subgraph polytope. Mathematical Programming, 33 (1985) 28–42CrossRefGoogle Scholar
  3. 3.
    Grötschel, M., Pulleyblank, W.: Weakly bipartite graphs and the max-cut problem. Operations Research Letters, 1 (1981) 23–27CrossRefGoogle Scholar
  4. 4.
    Hackbusch, W.: On the feedback vertex set problem for a planar graph. Technical report (1994). Institut für Informatik und Praktische Mathematik, Christian-Albrechts-Universität zu Kiel, D-24098 Kiel, GermanyGoogle Scholar
  5. 5.
    Jünger, M.: Polyhedral Combinatorics and the acyclic subdigraph problem., Research and exposition in mathematics 7 (1985). Heldermann Verlag, BerlinGoogle Scholar
  6. 6.
    Jünger, M., Reinelt, G., Thienel, S.: Practical problem solving with cutting plane algorithms in combinatorial optimization. In Cook, W., Lovász, L., Seymour, P., editors, Combinatorial Optimization, DIMACS series in Discrete Mathematics and Theoretical Computer Science 20 (1995) 111–152Google Scholar
  7. 7.
    Karp, R.: Reducibility among combinatorial problems. In Miller, R., Thatcher, J., editors, Complexity of Computer Computations, (1971) 85–103. Plenum Press, New YorkGoogle Scholar
  8. 8.
    Kevorkian, A.: General topological results on the construction of a minimum essential set of a directed graph. IEEE Transactions on Circuits and Systems, CAS-27(4) (1980) 293–304CrossRefGoogle Scholar
  9. 9.
    Kunzmann, A. Wunderlich, H.: An analytical approach to the partial scan problem. Journal of electronic testing: Theory and Applications, 1 (1990) 163–174Google Scholar
  10. 10.
    Lovász, L. Randomized Algorithms in Combinatorial Optimization. In Cook, W., Lovasz, L., Seymour, P., editors, Combinatorial Optimization, DIMACS series in Discrete Mathematics and Theoretical Computer Science 20 (1995) 153–179Google Scholar
  11. 11.
    Monien, B. Schulz, R.: Pour approximation algorithms for the feedback vertex set problem. In Graphtheoretic concepts in computer science, Proc. 7th Conf. Linz/Austria (1985) 315–326Google Scholar
  12. 12.
    Rosen, B.: Robust linear algorithms for cutsets. Journal of algorithms, 3 (1982) 205–217CrossRefGoogle Scholar
  13. 13.
    Smith, W., Walford, R.: The identification of a minimal feedback vertex set of a directed graph. IEEE Transactions on circuits and systems, CAS-22(1) (1975) 9–15CrossRefGoogle Scholar
  14. 14.
    Speckenmeyer, E.: Untersuchungen zum Feedback Vertex Set Problem in ungerichteten Graphen. Theoretische Informatik Bericht 16 (1983). Universität PaderbornGoogle Scholar
  15. 15.
    Stamm, H.: On feedback problems in planar digraphs. Graph-theoretic concepts in computer science, Proc. Int. Workshop, Berlin/Germany 1990, Lect. Notes Comput. Sci. 484 (1992) 79–89. Springer VerlagGoogle Scholar
  16. 16.
    Thienel, S.: ABACUS — A Branch And CUt System. PhD thesis (1995). Angewandte Mathematik und Informatik, Universität zu KölnGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Meinrad Funke
    • 1
  • Gerhard Reinelt
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HeidelbergHeidelberg

Personalised recommendations