Algorithmica

, Volume 64, Issue 1, pp 85–111 | Cite as

Parameterized Complexity of the Spanning Tree Congestion Problem

  • Hans L. Bodlaender
  • Fedor V. Fomin
  • Petr A. Golovach
  • Yota Otachi
  • Erik Jan van Leeuwen
Open Access
Article

Abstract

We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k≥8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k≤3 the problem becomes polynomially time solvable.

Keywords

Spanning tree congestion Graph minor Parameterized algorithms Apex graph 

References

  1. 1.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Borie, R.B., Parker, R.G., Tovey, C.A.: Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica 7, 555–581 (1992) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brandstädt, A., Dragan, F.F., Le, H.-O., Le, V.B.: Tree spanners on chordal graphs: complexity and algorithms. Theor. Comput. Sci. 310, 329–354 (2004) MATHCrossRefGoogle Scholar
  4. 4.
    Brandstädt, A., Dragan, F.F., Le, H.-O., Le, V.B., Uehara, R.: Tree spanners for bipartite graphs and probe interval graphs. Algorithmica 47, 27–51 (2007) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cai, L., Corneil, D.G.: Tree spanners. SIAM J. Discrete Math. 8, 359–387 (1995) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Castejón, A., Ostrovskii, M.I.: Minimum congestion spanning trees of grids and discrete toruses. Discuss. Math., Graph Theory 29, 511–519 (2009) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Cormen, T., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009) MATHGoogle Scholar
  8. 8.
    Courcelle, B.: The monadic second-order logic of graphs III: Tree-decompositions, minor and complexity issues. Theor. Inform. Appl. 26, 257–286 (1992) MathSciNetMATHGoogle Scholar
  9. 9.
    Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    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, 866–893 (2005) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Bidimensional parameters and local treewidth. SIAM J. Discrete Math. 18(3), 501–511 (2004) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Demaine, E.D., Hajiaghayi, M.: Graphs excluding a fixed minor have grids as large as treewidth,with combinatorial and algorithmic applications through bidimensionality. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), pp. 682–689. SIAM, Philadelphia (2005) Google Scholar
  13. 13.
    Demaine, E.D., Hajiaghayi, M.: Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica 28, 19–36 (2008) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Demaine, E.D., Hajiaghayi, M.T., Kawarabayashi, K.: Algorithmic graph minor theory: Decomposition, approximation, and coloring. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pp. 637–646. IEEE Computer Society, Los Alamitos (2005) CrossRefGoogle Scholar
  15. 15.
    Demaine, E.D., Hajiaghayi, M.T., Kawarabayashi, K.: Approximation algorithms via structural results for apex-minor-free graphs. In: Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP 2009). Lecture Notes in Computer Science, vol. 5555, pp. 316–327. Springer, Berlin (2009) Google Scholar
  16. 16.
    Diestel, R.: Graph Theory, 3rd edn. Springer, Berlin (2005) MATHGoogle Scholar
  17. 17.
    Dragan, F.F., Fomin, F.V., Golovach, P.A.: Spanners in sparse graphs, Journal of Computer and System Sciences, doi: 10.1016/j.jcss.2010.10.002
  18. 18.
    Dragan, F.F., Fomin, F.V., Golovach, P.A.: Approximation of minimum weight spanners for sparse graphs. Theor. Comput. Sci. 412, 846–852 (2011) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999) CrossRefGoogle Scholar
  20. 20.
    Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica 27, 275–291 (2000) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Fekete, S.P., Kremer, J.: Tree spanners in planar graphs. Discrete Appl. Math. 108, 85–103 (2001) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Fomin, F.V., Golovach, P.A., van Leeuwen, E.J.: Spanners of bounded degree graphs. Inf. Process. Lett. 111, 142–144 (2011) CrossRefGoogle Scholar
  23. 23.
    Geelen, J.F., Richter, R.B., Salazar, G.: Embedding grids in surfaces. Eur. J. Comb. 25, 785–792 (2004) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Grohe, M.: Local tree-width, excluded minors, and approximation algorithms. Combinatorica 23, 613–632 (2003) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Grohe, M., Marx, D.: On tree width, bramble size, and expansion. J. Comb. Theory, Ser. B 99, 218–228 (2009) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21, 549–568 (1974) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Hruska, S.W.: On tree congestion of graphs. Discrete Math. 308, 1801–1809 (2008) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Kozawa, K., Otachi, Y., Yamazaki, K.: On spanning tree congestion of graphs. Discrete Math. 309, 4215–4224 (2009) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Appl. Math. 52, 233–252 (1994) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Law, H.-F.: Spanning tree congestion of the hypercube. Discrete Math. 309, 6644–6648 (2009) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Lawler, E.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1976) MATHGoogle Scholar
  32. 32.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11, 329–343 (1982) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Löwenstein, C., Rautenbach, D., Regen, F.: On spanning tree congestion. Discrete Math. 309, 4653–4655 (2009) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    MacLane, S.: A combinatorial condition for planar graphs. Fundam. Math. 28, 22–32 (1937) MATHGoogle Scholar
  35. 35.
    Mohar, B.: Combinatorial local planarity and the width of graph embeddings. Can. J. Math. 44, 1272–1288 (1992) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (2001) MATHGoogle Scholar
  37. 37.
    Okamoto, Y., Otachi, Y., Uehara, R., Uno, T.: Hardness results and an exact exponential algorithm for the spanning tree congestion problem. In: Proceedings of the 8th Annual Conference on Theory and Applications of Models of Computation (TAMC 2011). Lecture Notes in Comput. Sci., vol. 6648, pp. 452–462 (2011) Google Scholar
  38. 38.
    Ostrovskii, M.I.: Minimal congestion trees. Discrete Math. 285, 219–226 (2004) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Ostrovskii, M.I.: Minimum congestion spanning trees in planar graphs. Discrete Math. 310, 1204–1209 (2010) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Otachi, Y., Bodlaender, H.L., van Leeuwen, E.J.: Complexity results for the spanning tree congestion problem. In: Proceedings of the 6th International Workshop on Graph Theoretic Concepts in Computer Science (WG 2010). Lecture Notes in Comput. Sci., vol. 6410, pp. 3–14 (2010) Google Scholar
  41. 41.
    Peleg, D.: Low stretch spanning trees. In: Proceedings of the 27th International Symposium on Mathematical Foundations of Computer Science (MFCS 2002). Lecture Notes in Comput. Sci., vol. 2420, pp. 68–80 (2002) CrossRefGoogle Scholar
  42. 42.
    Raspaud, A., Sýkora, O., Vrťo, I.: Congestion and dilation, similarities and differences: a survey. In: Proceedings of the 7th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2000), pp. 269–280. Carleton Scientific, Oxford (2000) Google Scholar
  43. 43.
    Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory, Ser. B 52, 153–190 (1991) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Robertson, N., Seymour, P.D.: Graph minors. XVI. Excluding a non-planar graph. J. Comb. Theory, Ser. B 89, 43–76 (2003) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Comb. Theory, Ser. B 62, 323–348 (1994) MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Simonson, S.: A variation on the min cut linear arrangement problem. Math. Syst. Theory 20, 235–252 (1987) MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Thomassen, C.: A simpler proof of the excluded minor theorem for higher surfaces. J. Comb. Theory, Ser. B 70, 306–311 (1997) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Hans L. Bodlaender
    • 1
  • Fedor V. Fomin
    • 2
  • Petr A. Golovach
    • 3
  • Yota Otachi
    • 4
  • Erik Jan van Leeuwen
    • 2
  1. 1.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
  2. 2.Department of InformaticsUniversity of BergenBergenNorway
  3. 3.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  4. 4.Graduate School of Information SciencesTohoku UniversitySendaiJapan

Personalised recommendations