# Near-Linear Time Constant-Factor Approximation Algorithm for Branch-Decomposition of Planar Graphs

• Qian-Ping Gu
• Gengchun Xu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

## Abstract

We give constant-factor approximation algorithms for branch-decomposition of planar graphs. Our main result is an algorithm which for an input planar graph $$G$$ of $$n$$ vertices and integer $$k$$, in $$O(n\log ^4n)$$ time either constructs a branch-decomposition of $$G$$ with width at most $$(2+\delta )k$$, $$\delta >0$$ is a constant, or a $$(k+1)\times \lceil {\frac{k+1}{2}}\rceil$$ cylinder minor of $$G$$ implying $${\mathrm{bw}}(G)>k$$, $${\mathrm{bw}}(G)$$ is the branchwidth of $$G$$. This is the first $$\tilde{O}(n)$$ time constant-factor approximation for branchwidth/treewidth and largest grid/cylinder minors of planar graphs and improves the previous $$\min \{O(n^{1+\epsilon }),O(nk^3)\}$$ ($$\epsilon >0$$ is a constant) time constant-factor approximations. For a planar graph $$G$$ and $$k={\mathrm{bw}}(G)$$, a branch-decomposition of width at most $$(2+\delta )k$$ and a $$g\times \frac{g}{2}$$ cylinder/grid minor with $$g=\frac{k}{\beta }$$, $$\beta >2$$ is constant, can be computed by our algorithm in $$O(n\log ^4n\log k)$$ time.

## Keywords

Branch-/tree-decompositions Grid minor Planar graphs Approximation algorithm

## References

1. 1.
Arnborg, S., Cornell, D., Proskurowski, A.: Complexity of finding embedding in a k-tree. SIAM J. Discrete Math. 8, 277–284 (1987)
2. 2.
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)
3. 3.
Bian, Z., Gu, Q.P., Marjan, M., Tamaki, H., Yoshitake, Y.: Empirical study on branchwidth and branch decomposition of planar graphs. In: Proceedings of Algorithm Engineering and Experimentation (ALENEX2008), pp. 152–165 (2008)Google Scholar
4. 4.
Bian, Z., Gu, Q.-P.: Computing branch decomposition of large planar graphs. In: McGeoch, C.C. (ed.) WEA 2008. LNCS, vol. 5038, pp. 87–100. Springer, Heidelberg (2008)
5. 5.
Bodlaender, H.L., Drange, P.G., Dreg, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M. : An $$O(c^kn)$$ 5-approximation algorithm for treewidth. In: Proceedings of the 2013 Annual Symposium on Foundation of Computer Science, (FOCS2013), pp. 499–508 (2013)Google Scholar
6. 6.
Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybern. 11, 1–21 (1993)
7. 7.
Bodlaender, H.L.: A linear time algorithm for finding tree-decomposition of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)
8. 8.
Bodlaender, H.L., Grigoriev, A., Koster, A.M.C.A.: Treewidth lower bounds with brambles. Algorithmica 51(1), 81–98 (2008)
9. 9.
Bodlaender, H.L., Thilikos, D.M.: Constructive linear time algorithm for branchwidth. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 627–637. Springer, Heidelberg (1997)
10. 10.
Borradaile, G., Sankowski, P., Wulff-Nilsen, C.: Min $$st$$-cut oracle for planar graphs with near-linear time preprocessing time. In: Proceedings of the 51st Annual Symposium on Foundations of Computer Science (FOCS2010), pp. 601–610 (2010) (also arXiv:1003.1320v2, April 2010)
11. 11.
Borradaile, G., Sankowski, P., Wulff-Nilsen, C.: Min $$st$$-cut oracle for planar graphs with near-linear time preprocessing time. arXiv:1003.1320v4 Oct 2013 (to appear in ACM TALG)
12. 12.
Demaine, E.D., Hajiaghayi, M.T.: Graphs excluding a fixed minor have grids as large as treewidth, with combinatorial and algorithmic applications through bidimensionality. In: Proceedings of the 2005 Symposium on Discrete Algorithms (SODA 2005), pp. 682–689 (2005)Google Scholar
13. 13.
Dorn, F., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Subexponential parameterized algorithms on bounded-genus graphs and $$H$$-minor-free graphs. J. of ACM 52(6), 866–893 (2005)
14. 14.
Fakcharoenphol, J., Rao, S.: Planar graphs, negative weight edges, shortest paths, and near linear time. J. Comput. Syst. Sci. 72, 868–889 (2006)
15. 15.
Feige, U., Hajiaghayi, M.T., Lee, J.R.: Improved approximation algorithms for minimum weight vertex separators. SIAM J. Comput. 38(2), 629–657 (2008)
16. 16.
Grigoriev, A.: Tree-width and large grid minors in planar graphs. Discrete Math. Theor. Comput. Sci. 13(1), 13–20 (2011)
17. 17.
Gu, Q.P., Tamaki, H.: Optimal branch decomposition of planar graphs in $${O}(n^3)$$ time. ACM Trans. Algorithms 4(3), Article No. 30, 1–13 (2008)Google Scholar
18. 18.
Gu, Q.P., Tamaki, H.: Constant-factor approximations of branch-decomposition and largest grid minor of planar graphs in $$O(n^{1+\epsilon })$$ time. Theor. Comput. Sci. 412, 4100–4109 (2011)
19. 19.
Gu, Q.P., Tamaki, H.: Improved bound on the planar branchwidth with respect to the largest grid minor size. Algorithmica 64, 416–453 (2012)
20. 20.
Gu, Q.P., Xu, G.: Near-linear time constant-factor approximation algorithm for branch-decomposition of planar graphs. arXiv:1407.6761, July 2014
21. 21.
Hicks, I.V.: Planar branch decompositions I: the ratcatcher. INFORMS J. Comput. 17(4), 402–412 (2005)
22. 22.
Hicks, I.V.: Planar branch decompositions II: the cycle method. INFORMS J. Comput. 17(4), 413–421 (2005)
23. 23.
Klein, P.N., Mozes, S., Sommer, C.: Structural recursive separator decompositions for planar graphs in linear time. In: Proceedings of the 2013 Annual ACM Symposium on the Theory of Computing (STOC2013), pp. 505–514 (2013)Google Scholar
24. 24.
Kammer, F., Tholey, T.: Approximate tree decompositions of planar graphs in linear time. In: Proceedings of the 2012 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA2012), pp. 683–698 (2012)Google Scholar
25. 25.
Kammer, F., Tholey, T.: Approximate tree decompositions of planar graphs in linear time. arXiv:1104.2275v2, May 2013
26. 26.
Mozes, S., Sommer, C.: Exact distance oracles for planar graphs. In: Proceedings of the 2012 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2012), pp. 209–222 (2012)Google Scholar
27. 27.
Robertson, N., Seymour, P.D.: Graph minors X. Obstructions to tree decomposition. J. Comb. Theory, Ser. B 52, 153–190 (1991)
28. 28.
Robertson, N., Seymour, P.D.: Graph minors XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63, 65–110 (1995)
29. 29.
Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Comb. Theory, Ser. B 62, 323–348 (1994)
30. 30.
Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)
31. 31.
Smith, J.C., Ulusal, E., Hicks, I.V.: A combinatorial optimization algorithm for solving the branchwidth problem. Comput. Optim. Appl. 51(3), 1211–1229 (2012)
32. 32.
Tamaki, H.: A linear time heuristic for the branch-decomposition of planar graphs. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 765–775. Springer, Heidelberg (2003)