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

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 n) time either constructs a branch-decomposition of G with width at most (2 + δ)k, δ > 0 is a constant, or a (k + 1)× ⌈ 2 ⌉ cylinder minor of G implying bw(G) > k, bw(G) is the branchwidth of G. This is the first Õ(n) time constantfactor approximation for branchwidth/treewidth and largest grid/cylinder minors of planar graphs and improves the previous min{O(n1+ǫ), O(nk3)} (ǫ > 0 is a constant) time constantfactor approximations. For a planar graph G and k = bw(G), a branch-decomposition of width at most (2 + δ)k and a g × g 2 cylinder/grid minor with g = k β , β > 2 is constant, can be computed by our algorithm in O(n log n log k) time.