Document Open Access Logo

The Parameterized Complexity of the Minimum Shared Edges Problem

Authors Till Fluschnik, Stefan Kratsch, Rolf Niedermeier, Manuel Sorge



PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2015.448.pdf
  • Filesize: 0.49 MB
  • 15 pages

Document Identifiers

Author Details

Till Fluschnik
Stefan Kratsch
Rolf Niedermeier
Manuel Sorge

Cite AsGet BibTex

Till Fluschnik, Stefan Kratsch, Rolf Niedermeier, and Manuel Sorge. The Parameterized Complexity of the Minimum Shared Edges Problem. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 448-462, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.FSTTCS.2015.448

Abstract

We study the NP-complete Minimum Shared Edges (MSE) problem. Given an undirected graph, a source and a sink vertex, and two integers p and k, the question is whether there are p paths in the graph connecting the source with the sink and sharing at most k edges. Herein, an edge is shared if it appears in at least two paths. We show that MSE is W[1]-hard when parameterized by the treewidth of the input graph and the number k of shared edges combined. We show that MSE is fixed-parameter tractable with respect to p, but does not admit a polynomial-size kernel (unless NP is a subset of coNP/poly). In the proof of the fixed-parameter tractability of MSE parameterized by p, we employ the treewidth reduction technique due to Marx, O'Sullivan, and Razgon [ACM TALG 2013].
Keywords
  • Parameterized complexity
  • kernelization
  • treewidth
  • treewidth reduction

Metrics

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

References

  1. Yusuke Aoki, Bjarni V. Halldórsson, Magnús M. Halldórsson, Takehiro Ito, Christian Konrad, and Xiao Zhou. The minimum vulnerability problem on graphs. In Proc. 8th International Conference on Combinatorial Optimization and Applications (COCOA'14), volume 8881 of LNCS, pages 299-313. Springer, 2014. Google Scholar
  2. Sepehr Assadi, Ehsan Emamjomeh-Zadeh, Ashkan Norouzi-Fard, Sadra Yazdanbod, and Hamid Zarrabi-Zadeh. The minimum vulnerability problem. In Proc. 23rd International Symposium on Algorithms and Computation (ISAAC'12), volume 7676 of LNCS, pages 382-391. Springer, 2012. Google Scholar
  3. René van Bevern, Andreas Emil Feldmann, Manuel Sorge, and Ondřej Suchý. On the parameterized complexity of computing balanced partitions in graphs. Theory of Computing Systems, 57(1):1-35, 2015. Google Scholar
  4. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM Journal on Discrete Mathematics, 28(1):277-305, 2014. Google Scholar
  5. Rajesh Chitnis, László Egri, and Dániel Marx. List H-coloring a graph by removing few vertices. In Proc. 21st European Symposion on Algorithms (ESA '13), volume 8125 of LNCS, pages 313-324. Springer, 2013. Google Scholar
  6. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  7. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Proc. IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS'11), pages 150-159, 2011. Google Scholar
  8. Reinhard Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer, 4th edition, 2010. Google Scholar
  9. Apostolos Dimitromanolakis. Analysis of the Golomb ruler and the Sidon set problems, and determination of large, near-optimal Golomb rulers. Master’s thesis, Department of Electronic and Computer Engineering, Technical University of Crete, June 2002. URL: http://www.cs.toronto.edu/~apostol/golomb/.
  10. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  11. Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science, 410(1):53-61, 2009. Google Scholar
  12. Michael R. Fellows, Bart M. P. Jansen, and Frances A. Rosamond. Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity. European Journal of Combinatorics, 34(3):541-566, 2013. Google Scholar
  13. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006. Google Scholar
  14. Till Fluschnik. The parameterized complexity of finding paths with shared edges. Master thesis, Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, 2015. URL: http://fpt.akt.tu-berlin.de/publications/theses/MA-till-fluschnik.pdf.
  15. Gregory Gutin, Mark Jones, and Bin Sheng. Parameterized complexity of the k-arc Chinese postman problem. In Proc. 22nd European Symposium on Algorithms (ESA '14), volume 8737 of LNCS, pages 530-541. Springer, 2014. Google Scholar
  16. Ton Kloks. Treewidth, Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer, 1994. Google Scholar
  17. Dániel Marx, Barry O'Sullivan, and Igor Razgon. Finding small separators in linear time via treewidth reduction. ACM Transactions on Algorithms, 9(4):30, 2013. Google Scholar
  18. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006. Google Scholar
  19. Rolf Niedermeier. Reflections on multivariate algorithmics and problem parameterization. In Proc. 27th International Symposium on Theoretical Aspects of Computer Science (STACS '10), volume 5 of LIPIcs, pages 17-32. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010. Google Scholar
  20. Masoud T. Omran, Jörg-Rüdiger Sack, and Hamid Zarrabi-Zadeh. Finding paths with minimum shared edges. Journal of Combinatorial Optimization, 26(4):709-722, 2013. Google Scholar
  21. Andrzej Pelc. Fault-tolerant broadcasting and gossiping in communication networks. Networks, 28(3):143-156, 1996. Google Scholar
  22. Neil Robertson and Paul D. Seymour. Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms, 7(3):309-322, 1986. Google Scholar
  23. Douglas B. West. Introduction to Graph Theory. Prentice Hall, 2 edition, 2000. Google Scholar
  24. Zhi-Qian Ye, Yi-Ming Li, Hui-Qiang Lu, and Xiao Zhou. Finding paths with minimum shared edges in graphs with bounded treewidths. In Proc. Frontiers of Computer Science (FCS'13), pages 40-46, 2013. 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