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The Firefighter Problem: A Structural Analysis

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Parameterized and Exact Computation (IPEC 2014)

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Abstract

We consider the complexity of the firefighter problem where a budget of \({b \ge 1}\) firefighters are available at each time step. This problem is proved to be NP-complete even on trees of degree at most three and \({b = 1}\) [10] and on trees of bounded degree \(b+3\) for any fixed \(b \ge 2\) [3]. In this paper, we provide further insight into the complexity landscape of the problem by showing a complexity dichotomy result with respect to the parameters pathwidth and maximum degree of the input graph. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any \(b \ge 1\). Then we show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter “pathwidth” and “maximum degree” of the input graph. Finally, we show that the problem remains NP-complete on very dense graphs, namely co-bipartite graphs, but is fixed-parameter tractable with respect to the parameter “cluster vertex deletion”.

Morgan Chopin—A major part of this work was done during a three-month visit of the University of Portsmouth supported by the ERASMUS program.

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References

  1. Anshelevich, E., Chakrabarty, D., Hate, A., Swamy, C.: Approximability of the firefighter problem. Algorithmica 62(1–2), 520–536 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bazgan, C., Chopin, M., Cygan, M., Fellows, M.R., Fomin, F.V., van Leeuwen, E.J.: Parameterized complexity of firefighting. J. Comput. Syst. Sci. 80(7), 1285–1297 (2014)

    Article  MATH  Google Scholar 

  3. Bazgan, C., Chopin, M., Ries, B.: The firefighter problem with more than one firefighter on trees. Discrete Appl. Math. 161(7–8), 899–908 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bodlaender, H.L.: Classes of graphs with bounded tree-width. Bull. EATCS 36, 116–128 (1988)

    MATH  Google Scholar 

  5. Cai, L., Verbin, E., Yang, L.: Firefighting on trees: (1 \(-\) 1/e)–approximation, fixed parameter tractability and a subexponential algorithm. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 258–269. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  6. Chung, F.R., Seymour, P.D.: Graphs with small bandwidth and cutwidth. In: Graph Theory and combinatorics 1988 Proceedings of the Cambridge Combinatorial Conference in Honour of Paul Erdös, Annals of Discrete Mathematics, vol. 43, pp. 113–119 (1989)

    Google Scholar 

  7. Costa, V., Dantas, S., Dourado, M.C., Penso, L., Rautenbach, D.: More fires and more fighters. Discrete Appl. Math. 161(16–17), 2410–2419 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Develin, M., Hartke, S.G.: Fire containment in grids of dimension three and higher. Discrete Appl. Math. 155(17), 2257–2268 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Doucha, M., Kratochvíl, J.: Cluster vertex deletion: a parameterization between vertex cover and clique-width. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 348–359. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  10. Finbow, S., King, A., MacGillivray, G., Rizzi, R.: The firefighter problem for graphs of maximum degree three. Discrete Math. 307(16), 2094–2105 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fomin, F.V., Heggernes, P., van Leeuwen, E.J.: Making Life Easier for Firefighters. In: Kranakis, E., Krizanc, D., Luccio, F. (eds.) FUN 2012. LNCS, vol. 7288, pp. 177–188. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H Freeman and Company, New York (1979)

    MATH  Google Scholar 

  13. Hartnell, B.: Firefighter! an application of domination, Presentation. In: 10th Conference on Numerical Mathematics and Computing, University of Manitoba in Winnipeg, Canada (1995)

    Google Scholar 

  14. Hartnell, B., Li, Q.: Firefighting on trees: how bad is the greedy algorithm? Congressus Numerantium 145, 187–192 (2000)

    MATH  MathSciNet  Google Scholar 

  15. Iwaikawa, Y., Kamiyama, N., Matsui, T.: Improved approximation algorithms for firefighter problem on trees. IEICE Trans. Inf. Syst. E94.D(2), 196–199 (2011)

    Article  Google Scholar 

  16. King, A., MacGillivray, G.: The firefighter problem for cubic graphs. Discrete Math. 310(3), 614–621 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Korach, E., Solel, N.: Tree-width, path-width, and cutwidth. Discrete Appl. Math. 43(1), 97–101 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. MacGillivray, G., Wang, P.: On the firefighter problem. J. Comb. Math. Comb. Comput. 47, 83–96 (2003)

    MATH  MathSciNet  Google Scholar 

  19. Ng, K.L., Raff, P.: A generalization of the firefighter problem on ZxZ. Discrete Appl. Math. 156(5), 730–745 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Portsmouth, School of Computing, Portsmouth, UK

    Janka Chlebíková

  2. Institut für Optimierung und Operations Research, Universität Ulm, Ulm, Germany

    Morgan Chopin

Authors
  1. Janka Chlebíková
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  2. Morgan Chopin
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Correspondence to Morgan Chopin .

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Editors and Affiliations

  1. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

    Marek Cygan

  2. University of Bergen, Bergen, Norway

    Pinar Heggernes

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Chlebíková, J., Chopin, M. (2014). The Firefighter Problem: A Structural Analysis. In: Cygan, M., Heggernes, P. (eds) Parameterized and Exact Computation. IPEC 2014. Lecture Notes in Computer Science(), vol 8894. Springer, Cham. https://doi.org/10.1007/978-3-319-13524-3_15

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  • DOI: https://doi.org/10.1007/978-3-319-13524-3_15

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-13524-3

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