We give constant-factor approximation algorithms for computing the optimal branch-decompositions and largest grid minors of planar graphs. For a planar graph with vertices, let be the branchwidth of and the largest integer such that has a grid as a minor. Let be a fixed integer and arbitrary constants satisfying and . We give an algorithm which constructs in time a branch-decomposition of with width at most . We also give an algorithm which constructs a grid minor of with in time. The constants hidden in the Big-O notations are proportional to and , respectively.
A preliminary version of this paper appeared in (Q.P. Gu, H. Tamaki, Constant-factor approximations of branch-decomposition and largest grid minor of planar graphs in time, in: Proc. of the 2009 International Symposium on Algorithms and Computation, ISAAC 2009, 2009, pp. 984–993) [19].