Parameterized Computational Geometry via Decomposition Theorems

  • Fahad Panolan
  • Saket SaurabhEmail author
  • Meirav Zehavi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)


Parameterized complexity is one of the most established algorithmic paradigms to deal with computationally hard problems. In the first two decades, the field largely focused on problems arising from studies of graphs and networks. However, lately the focus has changed substantially and it has started to permeate into other fields such as computational geometry, and computational social choice theory. In this article, we will survey some exciting developments in the emerging field of parameterized computational geometry through our contributions. We will focus on designing efficient parameterized algorithms on unit-disk graphs via new graph decomposition theorems.


  1. 1.
    Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial-time data reduction for dominating set. J. ACM 51(3), 363–384 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (meta) kernelization. J. ACM 63(5), 44:1–44:69 (2016). Scholar
  5. 5.
    Bui, T.N., Peck, A.: Partitioning planar graphs. SIAM J. Comput. 21(2), 203–215 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chen, J., Fernau, H., Kanj, I.A., Xia, G.: Parametric duality and kernelization: lower bounds and upper bounds on kernel size. SIAM J. Comput. 37(4), 1077–1106 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discret. Math. 86(1–3), 165–177 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). Scholar
  9. 9.
    Cygan, M., Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Minimum bisection is fixed parameter tractable. In: Symposium on Theory of Computing, STOC 2014, New York, NY, USA, 31 May–03 June 2014, pp. 323–332 (2014)Google Scholar
  10. 10.
    Dawar, A., Grohe, M., Kreutzer, S., Schweikardt, N.: Approximation schemes for first-order definable optimisation problems. In: LICS 2006, pp. 411–420 (2006)Google Scholar
  11. 11.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded genus and \(H\)-minor-free graphs. J. ACM 52(6), 866–893 (2005)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Demaine, E.D., Hajiaghayi, M.: Bidimensionality. In: Kao, M.Y. (ed.) Encyclo-pedia of Algorithms. Springer, Boston (2008). Scholar
  13. 13.
    Demaine, E.D., Hajiaghayi, M.: The bidimensionality theory and its algorithmic applications. Comput. J. 51(3), 292–302 (2008)CrossRefGoogle Scholar
  14. 14.
    Demaine, E.D., Hajiaghayi, M., Mohar, B.: Approximation algorithms via contraction decomposition. Combinatorica 30(5), 533–552 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  16. 16.
    Dorn, F., Fomin, F.V., Thilikos, D.M.: Subexponential parameterized algorithms. Comput. Sci. Rev. 2(1), 29–39 (2008)zbMATHCrossRefGoogle Scholar
  17. 17.
    Eisenstat, D., Klein, P., Mathieu, C.: An efficient polynomial-time approximation scheme for Steiner forest in planar graphs. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 626–638. SIAM (2012)Google Scholar
  18. 18.
    Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica 27, 275–291 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Fahad Panolan, S.S., Zehavi, M.: Contraction decomposition in unit disk graphs and algorithmic applications in parameterized complexity. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2019, to appear)Google Scholar
  20. 20.
    Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S., Zehavi, M.: Finding, hitting and packing cycles in subexponential time on unit disk graphs. In: 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017. LIPIcs, 10–14 July 2017, Warsaw, Poland, vol. 80, pp. 65:1–65:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  21. 21.
    Fomin, F.V., Lokshtanov, D., Saurabh, S.: Excluded grid minors and efficient polynomial-time approximation schemes. J. ACM 65(2), 10:1–10:44 (2018). Scholar
  22. 22.
    Fomin, F.V., Thilikos, D.M.: New upper bounds on the decomposability of planar graphs. J. Graph Theory 51(1), 53–81 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. J. Algorithms 53(1), 55–84 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Grohe, M.: Local tree-width, excluded minors, and approximation algorithms. Combinatorica 23(4), 613–632 (2003). Scholar
  25. 25.
    Gu, Q.P., Tamaki, H.: Improved bounds on the planar branchwidth with respect to the largest grid minor size. Algorithmica 64(3), 416–453 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ito, H., Kadoshita, M.: Tractability and intractability of problems on unit disk graphs parameterized by domain area. In: Proceedings of the 9th International Symposium on Operations Research and Its Applications (ISORA 2010), pp. 120–127 (2010)Google Scholar
  27. 27.
    Khanna, S., Motwani, R.: Towards a syntactic characterization of PTAS. In: STOC 1996, pp. 329–337. ACM (1996)Google Scholar
  28. 28.
    Klein, P.N., Marx, D.: A subexponential parameterized algorithm for subset TSP on planar graphs. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1812–1830. SIAM (2014)Google Scholar
  29. 29.
    Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9, 615–627 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Marx, D.: The square root phenomenon in planar graphs. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7966, p. 28. Springer, Heidelberg (2013). Scholar
  31. 31.
    Raghavan, V., Spinrad, J.: Robust algorithms for restricted domains. J. Algorithms 48(1), 160–172 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Comb. Theory Ser. B 62(2), 323–348 (1994)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of BergenBergenNorway
  2. 2.The Institute of Mathematical Sciences, HBNIChennaiIndia
  3. 3.Ben-Gurion UniversityBeershebaIsrael

Personalised recommendations