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Large Independent Sets in Subquartic Planar Graphs

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WALCOM: Algorithms and Computation (WALCOM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9627))

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Abstract

By the famous Four Color Theorem, every planar graph admits an independent set that contains at least one quarter of its vertices. This lower bound is tight for infinitely many planar graphs, and finding maximum independent sets in planar graphs is \(\mathsf {NP}\)-hard. A well-known open question in the field of Parameterized Complexity asks whether the problem of finding a maximum independent set in a given planar graph is fixed-parameter tractable, for parameter the “gain” over this tight lower bound. This open problem has been posed many times [4, 8, 10, 13, 17, 20, 31, 32, 35, 38].

We show fixed-parameter tractability of the independent set problem parameterized above tight lower bound in planar graphs with maximum degree at most 4, in subexponential time.

This research was supported by ERC Starting Grant 306465 (BeyondWorstCase).

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References

  1. Albertson, M., Bollobas, B., Tucker, S.: The independence ratio and maximum degree of a graph. In: Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 43–50. Congressus Numerantium, No. XVII. Utilitas Math., Winnipeg, Man. (1976)

    Google Scholar 

  2. Appel, K., Haken, W.: Every planar map is four colorable. Bull. Amer. Math. Soc. 82(5), 711–712 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berge, C.: Graphs and Hypergraphs, revised edn. North-Holland Publishing Co., Amsterdam (1976)

    MATH  Google Scholar 

  4. Bodlaender, H.L.: Open problems in parameterized and exact computation. Technical report UU-CS-2008-017, Utrecht University (2008)

    Google Scholar 

  5. Brooks, R.L.: On colouring the nodes of a network. Proc. Camb. Philos. Soc. 37, 194–197 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. J. Comput. Syst. Sci. 67(4), 789–807 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cranston, D.W., Rabern, L.: Planar graphs are 9/2-colorable and have independence ratio at least 3/13 (2015). http://arxiv.org/abs/1410.7233

  8. Crowston, R., Fellows, M., Gutin, G., Jones, M., Rosamond, F., Thomassé, S., Yeo, A.: Simultaneously satisfying linear equations over \(\mathbb{F}_2\): MaxLin2 and Max-\(r\)-Lin2 parameterized above average. In: Proceedings of FSTTCS 2011, pp. 229–240 (2011)

    Google Scholar 

  9. Crowston, R., Jones, M., Mnich, M.: Max-cut parameterized above the Edwards-Erdős bound. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 242–253. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  10. Cygan, M., Fomin, F., Jansen, B., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Open problems from the Bedlewo school on parameterized algorithms and complexity (2014). http://fptschool.mimuw.edu.pl/opl.pdf

  11. Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, New York (2015)

    Book  MATH  Google Scholar 

  12. Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: On multiway cut parameterized above lower bounds. ACM Trans. Comput. Theory 5(1), 3:1–3:11 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dvořák, Z., Mnich, M.: Large independent sets in triangle-free planar graphs. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 346–357. Springer, Heidelberg (2014)

    Google Scholar 

  14. Dvořák, Z., Sereni, J.-S.S., Volec, J.: Subcubic triangle-free graphs have fractional chromatic number at most 14/5. J. London Math. Soc. 89(3), 641–662 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl. 3(3), 27 (electronic) (1999)

    Google Scholar 

  16. Faria, L., Klein, S., Stehlík, M.: Odd cycle transversals and independent sets in fullerene graphs. SIAM J. Discrete Math. 26(3), 1458–1469 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fellows, M.R., Guo, J., Marx, D., Saurabh, S.: Data reduction and problem kernels (Dagstuhl Seminar 12241). Dagstuhl Reports 2(6), 26–50 (2012)

    Google Scholar 

  18. Fleischner, H., Sabidussi, G., Sarvanov, V.I.: Maximum independent sets in 3- and 4-regular Hamiltonian graphs. Discrete Math. 310(20), 2742–2749 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, New York (1979)

    MATH  Google Scholar 

  20. Giannopoulou, A.C., Kolay, S., Saurabh, S.: New lower bound on Max Cut of hypergraphs with an application to r-Set Splitting. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 408–419. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  21. Grohe, M.: Local tree-width, excluded minors, and approximation algorithms. Combinatorica 23(4), 613–632 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Grötzsch, H.: Zur Theorie der diskreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe, 8, 109–120 (1958/1959)

    Google Scholar 

  23. Gutin, G., Jones, M., Yeo, A.: Kernels for below-upper-bound parameterizations of the hitting set and directed dominating set problems. Theoret. Comput. Sci. 412(41), 5744–5751 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gutin, G., Kim, E.J., Mnich, M., Yeo, A.: Betweenness parameterized above tight lower bound. J. Comput. System Sci. 76(8), 872–878 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gutin, G., van Iersel, L., Mnich, M., Yeo, A.: Every ternary permutation constraint satisfaction problem parameterized above average has a kernel with a quadratic number of variables. J. Comput. System Sci. 78(1), 151–163 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Heckman, C.C., Thomas, R.: Independent sets in triangle-free cubic planar graphs. J. Combin. Theory Ser. B 96(2), 253–275 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kammer, F., Tholey, T.: Approximate tree decompositions of planar graphs in linear time. In: Proceedings of SODA 2012, pp. 683–698 (2012)

    Google Scholar 

  28. King, A.D., Lu, L., Peng, X.: A fractional analogue of Brooks’ theorem. SIAM J. Discrete Math. 26(2), 452–471 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lu, L., Peng, X.: The fractional chromatic number of triangle-free graphs with \(\Delta \le 3\). Discrete Math. 312(24), 3502–3516 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31(2), 335–354 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mahajan, M., Raman, V., Sikdar, S.: Parameterizing above or below guaranteed values. J. Comput. System Sci. 75(2), 137–153 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mnich, M.: Algorithms in moderately exponential time. Ph.D. thesis, TU Eindhoven (2010)

    Google Scholar 

  33. Mnich, M., Zenklusen, R.: Bisections above tight lower bounds. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds.) WG 2012. LNCS, vol. 7551, pp. 184–193. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  34. Molloy, M., Reed, B.: Graph Colouring and the Probabilistic Method. Algorithms and Combinatorics, vol. 23. Springer, Berlin (2002)

    MATH  Google Scholar 

  35. Niedermeier, R.: Invitation to Fixed-parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. OUP, Oxford (2006)

    Book  MATH  Google Scholar 

  36. Robertson, N., Sanders, D., Seymour, P., Thomas, R.: The four-colour theorem. J. Combin. Theory Ser. B 70(1), 2–44 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Scheinerman, E.R., Ullman, D.H.: Fractional Graph Theory. Dover Publications Inc., Mineola (2011)

    MATH  Google Scholar 

  38. Sikdar, S.: Parameterizing from the extremes. Ph.D. thesis, The Institute of Mathematical Sciences, Chennai (2010)

    Google Scholar 

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Acknowledgements

I am indebted to Zdeněk Dvořák for helpful remarks, and an anonymous reviewer who suggested considering treewidth over pathwidth.

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  1. Universität Bonn, Bonn, Germany

    Matthias Mnich

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  1. Matthias Mnich
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Correspondence to Matthias Mnich .

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Editors and Affiliations

  1. Dept. of Computer Science and Engin., Bangladesh Univ. of Engin. & Technol., Dhaka, Bangladesh

    Mohammad Kaykobad

  2. Dept. of Computer Science, Sapienza University of Rome, Rome, Italy

    Rossella Petreschi

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Mnich, M. (2016). Large Independent Sets in Subquartic Planar Graphs. In: Kaykobad, M., Petreschi, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2016. Lecture Notes in Computer Science(), vol 9627. Springer, Cham. https://doi.org/10.1007/978-3-319-30139-6_17

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  • DOI: https://doi.org/10.1007/978-3-319-30139-6_17

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30138-9

  • Online ISBN: 978-3-319-30139-6

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