Document Open Access Logo

Polynomial Kernels for Deletion to Classes of Acyclic Digraphs

Authors Matthias Mnich, Erik Jan van Leeuwen

PDF
Thumbnail PDF

File

LIPIcs.STACS.2016.55.pdf
  • Filesize: 0.63 MB
  • 13 pages

Document Identifiers

Author Details

Matthias Mnich
Erik Jan van Leeuwen

Cite As Get BibTex

Matthias Mnich and Erik Jan van Leeuwen. Polynomial Kernels for Deletion to Classes of Acyclic Digraphs. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.STACS.2016.55

Abstract

We consider the problem to find a set X of vertices (or arcs) with |X| <= k in a given digraph G such that D = G-X is an acyclic digraph. In its generality, this is DIRECTED FEEDBACK VERTEX SET or DIRECTED FEEDBACK ARC SET respectively. The existence of a polynomial kernel for these problems is a notorious open problem in the field of kernelization, and little progress has been made. 

In this paper, we consider both deletion problems with an additional restriction on D, namely that D must be an out-forest, an out-tree, or a (directed) pumpkin. Our main results show that for each of these three restrictions the vertex deletion problem remains NP-hard, but we can obtain a kernel with k^{O(1)} vertices on general digraphs G. We also show that, in contrast to the vertex deletion problem, the arc deletion problem with each of the above restrictions can be solved in polynomial time.

Subject Classification

Keywords
  • directed feedback vertex/arc set
  • parameterized algorithms
  • kernels

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. F. N. Abu-Khzam. A kernelization algorithm for d-hitting set. J. Comput. System Sci., 76(7):524-531, 2010. URL: http://dx.doi.org/10.1016/j.jcss.2009.09.002.
  2. V. Bafna, P. Berman, and T. Fujito. A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discrete Math., 12(3):289-297, 1999. URL: http://dx.doi.org/10.1137/S0895480196305124.
  3. J. Bang-Jensen, A. Maddaloni, and S. Saurabh. Algorithms and kernels for feedback set problems in generalizations of tournaments. Algorithmica, to appear. Google Scholar
  4. J. Chen, Y. Liu, S. Lu, B. O'Sullivan, and I. Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM, 55(5):21:1-21:19, November 2008. URL: http://dx.doi.org/10.1145/1411509.1411511.
  5. M. Cygan, F. Fomin, B.M.P. Jansen, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Open problems for FPT school. Technical report, Warsaw University, 2014. URL: http://fptschool.mimuw.edu.pl/opl.pdf.
  6. M. Cygan, L. Kowalik, and M. Pilipczuk. Open problems from workshop on kernels. Technical report, Warsaw University, 2013. URL: http://worker2013.mimuw.edu.pl/slides/worker-opl.pdf.
  7. M. Dom, J. Guo, F. Hüffner, R. Niedermeier, and A. Truß. Fixed-parameter tractability results for feedback set problems in tournaments. J. Discrete Algorithms, 8(1):76-86, 2010. URL: http://dx.doi.org/10.1016/j.jda.2009.08.001.
  8. R.G. Downey and M.R. Fellows. Fundamentals of Parameterized Complexity Theory. Springer, 2013. Google Scholar
  9. G. Even, J. Naor, B. Schieber, and M. Sudan. Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica, 20(2):151-174, 1998. URL: http://dx.doi.org/10.1007/PL00009191.
  10. M.R. Fellows, J. Guo, D. Marx, and S. Saurabh. Data reduction and problem kernels (Dagstuhl seminar 12241). Technical Report 2(6), Dagstuhl Reports, 2012. Google Scholar
  11. J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, 2006. Google Scholar
  12. F.V. Fomin, S. Saurabh, and Y. Villanger. A polynomial kernel for proper interval vertex deletion. SIAM J. Discrete Math., 27(4):1964-1976, 2013. URL: http://dx.doi.org/10.1137/12089051X.
  13. A.C. Giannopoulou, D. Lokshtanov, S. Saurabh, and O. Suchy. Tree deletion set has a polynomial kernel (but no OPT^O(1) approximation). In Proc. FSTTCS 2014, volume 29 of LIPIcs, pages 85-96, 2014. Google Scholar
  14. R.M. Karp. Reducibility among combinatorial problems. In Complexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), pages 85-103, New York, 1972. Plenum. Google Scholar
  15. D.G. Kirkpatrick and P. Hell. On the completeness of a generalized matching problem. In Proc. STOC 1978, pages 240-245, New York, 1978. ACM. Google Scholar
  16. A. Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003. Google Scholar
  17. P.D. Seymour. Packing directed circuits fractionally. Combinatorica, 15(2):281-288, 1995. URL: http://dx.doi.org/10.1007/BF01200760.
  18. E. Speckenmeyer. On feedback problems in digraphs. In Manfred Nagl, editor, Proc. WG 1990, volume 411 of Lecture Notes Comput. Sci., pages 218-231. Springer, 1990. URL: http://dx.doi.org/10.1007/3-540-52292-1_16.
  19. S. Thomassé. A 4k² kernel for feedback vertex set. ACM Trans. Algorithms, 6(2), 2010. URL: http://dx.doi.org/10.1145/1721837.1721848.
  20. M. Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems. J. ACM, 26(4):618-630, 1979. URL: http://dx.doi.org/10.1145/322154.322157.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact