Skip to main content

Parameterized Complexity of the Spanning Tree Congestion Problem

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.

References

  1. 1.

    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Cai, L., Corneil, D.G.: Tree spanners. SIAM J. Discrete Math. 8, 359–387 (1995)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Cormen, T., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)

    MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    Chapter  Google 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)

    MATH  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)

    Book  Google Scholar 

  20. 20.

    Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica 27, 275–291 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Fekete, S.P., Kremer, J.: Tree spanners in planar graphs. Discrete Appl. Math. 108, 85–103 (2001)

    MathSciNet  MATH  Article  Google 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)

    Article  Google Scholar 

  23. 23.

    Geelen, J.F., Richter, R.B., Salazar, G.: Embedding grids in surfaces. Eur. J. Comb. 25, 785–792 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

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

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Grohe, M., Marx, D.: On tree width, bramble size, and expansion. J. Comb. Theory, Ser. B 99, 218–228 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21, 549–568 (1974)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Hruska, S.W.: On tree congestion of graphs. Discrete Math. 308, 1801–1809 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Kozawa, K., Otachi, Y., Yamazaki, K.: On spanning tree congestion of graphs. Discrete Math. 309, 4215–4224 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Appl. Math. 52, 233–252 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Law, H.-F.: Spanning tree congestion of the hypercube. Discrete Math. 309, 6644–6648 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Lawler, E.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1976)

    MATH  Google Scholar 

  32. 32.

    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11, 329–343 (1982)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Löwenstein, C., Rautenbach, D., Regen, F.: On spanning tree congestion. Discrete Math. 309, 4653–4655 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    MacLane, S.: A combinatorial condition for planar graphs. Fundam. Math. 28, 22–32 (1937)

    MATH  Google Scholar 

  35. 35.

    Mohar, B.: Combinatorial local planarity and the width of graph embeddings. Can. J. Math. 44, 1272–1288 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (2001)

    MATH  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Ostrovskii, M.I.: Minimum congestion spanning trees in planar graphs. Discrete Math. 310, 1204–1209 (2010)

    MathSciNet  MATH  Article  Google 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)

    Chapter  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Comb. Theory, Ser. B 62, 323–348 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Simonson, S.: A variation on the min cut linear arrangement problem. Math. Syst. Theory 20, 235–252 (1987)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Fedor V. Fomin.

Additional information

Extended abstract of some results in this paper appeared in the proceedings of WG 2010 [40].

Y. Otachi belongs to JSPS Research Fellow.

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Bodlaender, H.L., Fomin, F.V., Golovach, P.A. et al. Parameterized Complexity of the Spanning Tree Congestion Problem. Algorithmica 64, 85–111 (2012). https://doi.org/10.1007/s00453-011-9565-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-011-9565-7

Keywords

  • Spanning tree congestion
  • Graph minor
  • Parameterized algorithms
  • Apex graph