LATIN 2004: LATIN 2004: Theoretical Informatics pp 109-118

Bidimensional Parameters and Local Treewidth

• Erik D. Demaine
• Fedor V. Fomin
• Dimitrios M. Thilikos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)

Abstract

For several graph theoretic parameters such as vertex cover and dominating set, it is known that if their values are bounded by k then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixed-parameter algorithms on minor-closed graph classes such as planar graphs, single-crossing-minor-free graphs, and graphs of bounded genus. In this paper we examine the question whether similar bounds can be obtained for larger minor-closed graph classes, and for general families of parameters including all the parameters where such a behavior has been reported so far.

Given a graph parameter P, we say that a graph family $$\mathcal{F}$$ has the parameter-treewidth property for P if there is a function f(p) such that every graph $$G \in \mathcal{F}$$ with parameter at most p has treewidth at most f(p). We prove as our main result that, for a large family of parameters called contraction-bidimensional parameters, a minor-closed graph family $$\mathcal{F}$$ has the parameter-treewidth property if $$\mathcal{F}$$ has bounded local treewidth. We also show “if and only if” for some parameters, and thus this result is in some sense tight. In addition we show that, for a slightly smaller family of parameters called minor-bidimensional parameters, all minor-closed graph families $$\mathcal{F}$$ excluding some fixed graphs have the parameter-treewidth property. The bidimensional parameters include many domination and covering parameters such as vertex cover, feedback vertex set, dominating set, edge-dominating set, q-dominating set (for fixed q). We use these theorems to develop new fixed-parameter algorithms in these contexts.

Keywords

Planar Graph Vertex Cover Tree Decomposition Domination Number Apex Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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, 461–493 (2002)
2. 2.
Alber, J., Fan, H., Fellows, M., Fernau, R.H., Niedermeier, R.: Refined search tree technique for dominating set on planar graphs. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 111–122. Springer, Heidelberg (2001)
3. 3.
Amir, E.: Efficient approximation for triangulation of minimum treewidth. In: Uncertainty in Artificial Intelligence: Proceedings of the Seventeenth Conference (UAI 2001), pp. 7–15. Morgan Kaufmann Publishers, San Francisco (2001)Google Scholar
4. 4.
Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. Assoc. Comput. Mach. 41, 153–180 (1994)
5. 5.
Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernetica 11, 1–23 (1993)
6. 6.
Courcelle, B.: Graph rewriting: an algebraic and logic approach. In: Handbook of theoretical computer science, vol. B, pp. 193–242. Elsevier, Amsterdam (1990)Google Scholar
7. 7.
Chen, J., Kanj, I.A., Jia, W.: Vertex cover: further observations and further improvements. Journal of Algorithms 41(2), 280–301 (2001)
8. 8.
Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Fixed-Parameter Algorithms for the (k, r)-Center in Planar Graphs and Map Graphs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 829–844. Springer, Heidelberg (2003)
9. 9.
Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs. To appear in SODA 2004 (2004)Google Scholar
10. 10.
Demaine, E.D., Hajiaghayi, M.T.: Fixed Parameter Algorithms for Minor-Closed Graphs (of Locally Bounded Treewidth). To appear in SODA 2004 (2004)Google Scholar
11. 11.
Demaine, E.D., Hajiaghayi, M.T., Thilikos, D.M.: Exponential speedup of fixed parameter algorithms on K 3,3-minor-free or K5-minor-free graphs. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 262–273. Springer, Heidelberg (2002)
12. 12.
Diestel, R.: Graph theory, 2nd edn. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2000)Google Scholar
13. 13.
Diestel, R., Jensen, T.R., Gorbunov, K.Y., Thomassen, C.: Highly connected sets and the excluded grid theorem. J. Combin. Theory Ser. B 75, 61–73 (1999)
14. 14.
Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, New York (1999)Google Scholar
15. 15.
Ellis, J., Fan, H., Fellows, M.: The dominating set problem is fixed parameter tractable for graphs of bounded genus. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, pp. 180–189. Springer, Heidelberg (2002)
16. 16.
Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica 27, 275–291 (2000)
17. 17.
Flum, J., Grohe, M.: Fixed-parameter tractability, definability, and modelchecking. SIAM J. Comput. 31, 113–145 (2001)
18. 18.
Fomin, F.V., Thilikos, D.M.: Dominating sets in planar graphs: branch-Width and exponential speed-up. In: SODA 2003, pp. 168–177 (2003)Google Scholar
19. 19.
Fomin, F.V., Thilikos, D.M.: A Simple and Fast Approach for Solving Problems on Planar Graphs. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 56–67. Springer, Heidelberg (2004)
20. 20.
Fomin, F.V., Thilikos, D.M.: Dominating sets and local treewidth. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 221–229. Springer, Heidelberg (2003)
21. 21.
Frick, M., Grohe, M.: Deciding first-order properties of locally treedecomposable graphs. J. ACM 48, 1184–1206 (2001)
22. 22.
Grohe, M.: Local tree-width, excluded minors, and approximation algorithms. To appear in CombinatoricaGoogle Scholar
23. 23.
Kanj, I., Perković, L.: Improved parameterized algorithms for planar dominating set. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 399–410. Springer, Heidelberg (2002)
24. 24.
Chang, M.-S., Kloks, T., Lee, C.-M.: Maximum clique transversals. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 300–310. Springer, Heidelberg (2001)Google Scholar
25. 25.
Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7, 309–322 (1986)
26. 26.
Robertson, N., Seymour, P.D.: Graph minors. V. Excluding a planar graph. J. Comb. Theory Series B 41, 92–111 (1986)

Authors and Affiliations

• Erik D. Demaine
• 1
• Fedor V. Fomin
• 2