Document Open Access Logo

Tracking Paths in Planar Graphs

Authors David Eppstein, Michael T. Goodrich , James A. Liu, Pedro Matias

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2019.54.pdf
  • Filesize: 0.67 MB
  • 17 pages

Document Identifiers

Author Details

David Eppstein
  • Department of Computer Science, University of California, Irvine, USA
Michael T. Goodrich
  • Department of Computer Science, University of California, Irvine, USA
James A. Liu
  • Department of Computer Science, University of California, Irvine, USA
Pedro Matias
  • Department of Computer Science, University of California, Irvine, USA

Acknowledgements

We thank Nil Mamano for suggesting the problem of tracking paths on a graph.

Cite AsGet BibTex

David Eppstein, Michael T. Goodrich, James A. Liu, and Pedro Matias. Tracking Paths in Planar Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.54

Abstract

We consider the NP-complete problem of tracking paths in a graph, first introduced by Banik et al. [Banik et al., 2017]. Given an undirected graph with a source s and a destination t, find the smallest subset of vertices whose intersection with any s-t path results in a unique sequence. In this paper, we show that this problem remains NP-complete when the graph is planar and we give a 4-approximation algorithm in this setting. We also show, via Courcelle’s theorem, that it can be solved in linear time for graphs of bounded-clique width, when its clique decomposition is given in advance.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Approximation Algorithm
  • Courcelle’s Theorem
  • Clique-Width
  • Planar
  • 3-SAT
  • Graph Algorithms
  • NP-Hardness

Metrics

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

References

  1. Norman TJ Bailey. The Mathematical Theory of Infectious Diseases and its Applications. Charles Griffin & Company Ltd, High Wycombe, United Kingdom, 2nd edition, 1975. Google Scholar
  2. Aritra Banik and Pratibha Choudhary. Fixed-Parameter Tractable Algorithms for Tracking Set Problems. In B. S. Panda and Partha P. Goswami, editors, Algorithms and Discrete Applied Mathematics - 4th International Conference, CALDAM 2018, Guwahati, India, February 15-17, 2018, Proceedings, volume 10743 of Lecture Notes in Computer Science, pages 93-104. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-74180-2_8.
  3. Aritra Banik, Pratibha Choudhary, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. A polynomial sized kernel for tracking paths problem. In Latin American Symposium on Theoretical Informatics, volume 10807 of Lecture Notes in Computer Science, pages 94-107. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-77404-6_8.
  4. Aritra Banik, Matthew J Katz, Eli Packer, and Marina Simakov. Tracking paths. In International Conference on Algorithms and Complexity, volume 10236 of Lecture Notes in Computer Science, pages 67-79. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-57586-5_7.
  5. Umberto Bertele and Francesco Brioschi. Nonserial Dynamic Programming. Academic Press, 1972. URL: https://www.elsevier.com/books/nonserial-dynamic-programming/bertele/978-0-12-093450-8.
  6. Sania Bhatti and Jie Xu. Survey of target tracking protocols using wireless sensor network. In 2009 Fifth International Conference on Wireless and Mobile Communications, pages 110-115. IEEE, 2009. URL: https://doi.org/10.1109/ICWMC.2009.25.
  7. Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Tracking Routes in Communication Networks. In Keren Censor-Hillel and Michele Flammini, editors, Structural Information and Communication Complexity - 26th International Colloquium, SIROCCO 2019, L'Aquila, Italy, July 1-4, 2019, Proceedings, volume 11639 of Lecture Notes in Computer Science, pages 81-93. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-24922-9_6.
  8. Hans L Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing, 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  9. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  10. Bruno Courcelle and Irène Durand. Automata for the verification of monadic second-order graph properties. J. Applied Logic, 10(4):368-409, 2012. URL: https://doi.org/10.1016/j.jal.2011.07.001.
  11. Bruno Courcelle and Irène Durand. Computations by fly-automata beyond monadic second-order logic. Theor. Comput. Sci., 619:32-67, 2016. URL: https://doi.org/10.1016/j.tcs.2015.12.026.
  12. Bruno Courcelle and Joost Engelfriet. Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach, volume 138 of Encyclopedia of mathematics and its applications. Cambridge University Press, 2012. URL: http://www.cambridge.org/fr/knowledge/isbn/item5758776/?site_locale=fr_FR.
  13. Bruno Courcelle, Joost Engelfriet, and Grzegorz Rozenberg. Handle-rewriting hypergraph grammars. Journal of Computer and System Sciences, 46(2):218-270, 1993. URL: https://doi.org/10.1016/0022-0000(93)90004-G.
  14. Bruno Courcelle, Johann A Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
  15. Bruno Courcelle and Mohamed Mosbah. Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science, 109(1-2):49-82, 1993. URL: https://doi.org/10.1016/0304-3975(93)90064-Z.
  16. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1-3):77-114, 2000. URL: https://doi.org/10.1016/S0166-218X(99)00184-5.
  17. Kee-Tai Goh, Jeffery Cutter, Bee-Hoon Heng, Stefan Ma, Benjamin KW Koh, Cynthia Kwok, Cheong-Mui Toh, and Suok-Kai Chew. Epidemiology and control of SARS in Singapore. Annals of the Academy of Medicine, Singapore, 35(5):301, 2006. URL: http://www.ncbi.nlm.nih.gov/pubmed/16829997.
  18. Rahul Gupta and Samir R Das. Tracking moving targets in a smart sensor network. In 2003 IEEE 58th Vehicular Technology Conference. VTC 2003-Fall, volume 5, pages 3035-3039. IEEE, 2003. URL: https://doi.org/10.1109/VETECF.2003.1286181.
  19. Rudolf Halin. S-functions for graphs. Journal of Geometry, 8(1-2):171-186, 1976. URL: https://doi.org/10.1007/BF01917434.
  20. Donald E Knuth and Arvind Raghunathan. The problem of compatible representatives. SIAM Journal on Discrete Mathematics, 5(3):422-427, 1992. URL: https://doi.org/10.1137/0405033.
  21. David Lichtenstein. Planar formulae and their uses. SIAM Journal on Computing, 11(2):329-343, 1982. URL: https://doi.org/10.1137/0211025.
  22. Cristopher Moore and Mark EJ Newman. Epidemics and percolation in small-world networks. Physical Review E, 61(5):5678, 2000. URL: https://doi.org/10.1103/PhysRevE.61.5678.
  23. Mark EJ Newman. Spread of epidemic disease on networks. Physical Review E, 66(1):016128, 2002. URL: https://doi.org/10.1103/PhysRevE.66.016128.
  24. Sang-Il Oum. Approximating rank-width and clique-width quickly. ACM Transactions on Algorithms, 5(1):10, 2008. URL: https://doi.org/10.1145/1435375.1435385.
  25. Tao Peng, Christopher Leckie, and Kotagiri Ramamohanarao. Survey of network-based defense mechanisms countering the DoS and DDoS problems. ACM Computing Surveys, 39(1):3, 2007. URL: https://doi.org/10.1145/1216370.1216373.
  26. Neil Robertson and Paul D. Seymour. Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  27. Devavrat Shah and Tauhid Zaman. Rumors in a network: Who’s the culprit? IEEE Transactions on Information Theory, 57(8):5163-5181, 2011. URL: https://doi.org/10.1109/TIT.2011.2158885.
  28. Alex C Snoeren, Craig Partridge, Luis A Sanchez, Christine E Jones, Fabrice Tchakountio, Stephen T Kent, and W Timothy Strayer. Hash-based IP traceback. ACM SIGCOMM Computer Communication Review, 31(4):3-14, 2001. URL: https://doi.org/10.1145/383059.383060.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact