Document Open Access Logo

New Results on Directed Edge Dominating Set

Authors Rémy Belmonte, Tesshu Hanaka, Ioannis Katsikarelis, Eun Jung Kim, Michael Lampis

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2018.67.pdf
  • Filesize: 0.54 MB
  • 16 pages

Document Identifiers

Author Details

Rémy Belmonte
  • University of Electro-Communications, Chofu, Tokyo, 182-8585, Japan
Tesshu Hanaka
  • Department of Information and System Engineering, Chuo University, Tokyo, Japan
Ioannis Katsikarelis
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, 75016, Paris, France
Eun Jung Kim
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243 , LAMSADE, 75016, Paris, France
Michael Lampis
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243 , LAMSADE, 75016, Paris, France

Funding

Partially supported by JSPS and MAEDI under the Japan-France Integrated Action Program (SAKURA), Project GRAPA 38593YJ.
  • Kim, Eun Jung: The author was partially supported by the ANR grant "ESIGMA" (ANR-17-CE40-0028).
  • Lampis, Michael: The author was partially supported by the ANR grant "ESIGMA" (ANR-17-CE40-0028).

Cite As Get BibTex

Rémy Belmonte, Tesshu Hanaka, Ioannis Katsikarelis, Eun Jung Kim, and Michael Lampis. New Results on Directed Edge Dominating Set. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.MFCS.2018.67

Abstract

We study a family of generalizations of Edge Dominating Set on directed graphs called Directed (p,q)-Edge Dominating Set. In this problem an arc (u,v) is said to dominate itself, as well as all arcs which are at distance at most q from v, or at distance at most p to u.
First, we give significantly improved FPT algorithms for the two most important cases of the problem, (0,1)-dEDS and (1,1)-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that (p,q)-dEDS is FPT parameterized by p+q+tw, but W-hard parameterized just by tw, where tw is the treewidth of the underlying graph of the input.
We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of p,q, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case (p=q=1) which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Edge Dominating Set
  • Tournaments
  • Treewidth

Metrics

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

References

  1. Paola Alimonti and Viggo Kann. Some apx-completeness results for cubic graphs. Theor. Comput. Sci., 237(1-2):123-134, 2000. Google Scholar
  2. Brenda S. Baker. Approximation Algorithms for NP-Complete Problems on Planar Graphs. J. ACM, 41(1):153-180, 1994. Google Scholar
  3. Daniel Binkele-Raible and Henning Fernau. Enumerate and measure: Improving parameter budget management. In IPEC, volume 6478 of Lecture Notes in Computer Science, pages 38-49. Springer, 2010. Google Scholar
  4. Arindam Biswas, Varunkumar Jayapaul, Venkatesh Raman, and Srinivasa Rao Satti. The Complexity of Finding (Approximate Sized) Distance-d Dominating Set in Tournaments. In Frontiers in Algorithmics, pages 22-33, 2017. Google Scholar
  5. Hans L. Bodlaender, Pål Grønås Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michal Pilipczuk. A c^k n 5-approximation algorithm for treewidth. SIAM J. Comput., 45(2):317-378, 2016. Google Scholar
  6. Rodica Boliac, Kathie Cameron, and Vadim V. Lozin. On computing the dissociation number and the induced matching number of bipartite graphs. Ars Comb., 72, 2004. Google Scholar
  7. Glencora Borradaile and Hung Le. Optimal dynamic program for r-domination problems over tree decompositions. In IPEC, volume 63 of LIPIcs, pages 8:1-8:23. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  8. Jean Cardinal, Stefan Langerman, and Eythan Levy. Improved approximation bounds for edge dominating set in dense graphs. Theor. Comput. Sci., 410(8-10):949-957, 2009. Google Scholar
  9. Miroslav Chlebík and Janka Chlebíková. Approximation hardness of edge dominating set problems. Journal of Combinatorial Optimization, 11(3):279-290, 2006. Google Scholar
  10. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  11. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Fixed-parameter algorithms for (k, r)-center in planar graphs and map graphs. ACM Trans. Algorithms, 1(1):33-47, 2005. Google Scholar
  12. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In STOC, pages 624-633. ACM, 2014. Google Scholar
  13. Rodney G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness i: Basic results. SIAM J. Comput., 24(4):873-921, 1995. Google Scholar
  14. Rodney G. Downey and Michael R. Fellows. Parameterized Computational Feasibility. In Feasible Mathematics II, pages 219-244, 1995. Google Scholar
  15. David Eisenstat, Philip N. Klein, and Claire Mathieu. Approximating k-center in planar graphs. In SODA, pages 617-627. SIAM, 2014. Google Scholar
  16. Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci., 410(1):53-61, 2009. Google Scholar
  17. Henning Fernau. edge dominating set: Efficient Enumeration-Based Exact Algorithms. In Parameterized and Exact Computation, pages 142-153. Springer Berlin Heidelberg, 2006. Google Scholar
  18. Fedor V. Fomin, Serge Gaspers, Saket Saurabh, and Alexey A. Stepanov. On two techniques of combining branching and treewidth. Algorithmica, 54(2):181-207, 2009. Google Scholar
  19. Toshihiro Fujito and Hiroshi Nagamochi. A 2-approximation algorithm for the minimum weight edge dominating set problem. Discrete Applied Mathematics, 118(3):199-207, 2002. Google Scholar
  20. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co., 1979. Google Scholar
  21. Torben Hagerup. Kernels for edge dominating set: Simpler or smaller. In MFCS, volume 7464 of Lecture Notes in Computer Science, pages 491-502. Springer, 2012. Google Scholar
  22. Tesshu Hanaka, Naomi Nishimura, and Hirotaka Ono. On directed covering and domination problems. In ISAAC, volume 92 of LIPIcs, pages 45:1-45:12. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  23. Frank Harary and Robert Z Norman. Some properties of line digraphs. Rendiconti del Circolo Matematico di Palermo, 9(2):161-168, 1960. Google Scholar
  24. Joseph D. Horton and Kyriakos Kilakos. Minimum edge dominating sets. SIAM J. Discret. Math., 6(3):375-387, 1993. Google Scholar
  25. Ken Iwaide and Hiroshi Nagamochi. An improved algorithm for parameterized edge dominating set problem. J. Graph Algorithms Appl., 20(1):23-58, 2016. Google Scholar
  26. Stephan Kreutzer and Siamak Tazari. Directed nowhere dense classes of graphs. In SODA '12, pages 1552-1562, 2012. Google Scholar
  27. Dana Moshkovitz. The projection games conjecture and the np-hardness of ln n-approximating set-cover. Theory of Computing, 11:221-235, 2015. Google Scholar
  28. Yury L. Orlovich, Alexandre Dolgui, Gerd Finke, Valery S. Gordon, and Frank Werner. The complexity of dissociation set problems in graphs. Discrete Applied Mathematics, 159(13):1352-1366, 2011. Google Scholar
  29. Richard Schmied and Claus Viehmann. Approximating edge dominating set in dense graphs. Theor. Comput. Sci., 414(1):92-99, 2012. Google Scholar
  30. Mingyu Xiao, Ton Kloks, and Sheung-Hung Poon. New parameterized algorithms for the edge dominating set problem. Theor. Comput. Sci., 511:147-158, 2013. Google Scholar
  31. Mingyu Xiao and Shaowei Kou. Exact algorithms for the maximum dissociation set and minimum 3-path vertex cover problems. Theor. Comput. Sci., 657:86-97, 2017. Google Scholar
  32. Mingyu Xiao and Hiroshi Nagamochi. A refined exact algorithm for edge dominating set. Theor. Comput. Sci., 560:207-216, 2014. Google Scholar
  33. Mihalis Yannakakis. Node-deletion problems on bipartite graphs. SIAM J. Comput., 10(2):310-327, 1981. Google Scholar
  34. Mihalis Yannakakis and Fanica Gavril. Edge dominating sets in graphs. SIAM Journ. on Applied Mathematics, 38(3):364-372, 1980. Google Scholar
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