Near-Linear Time Constant-Factor Approximation Algorithm for   Branch-Decomposition of Planar Graphs
  We give an algorithm which for an input planar graph $G$ of $n$ vertices and integer $k$, in $\min\{O(n\log^3n),O(nk^2)\}$ 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 $bw(G)>k$, $bw(G)$ 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 $\min\{O(n^{1+\epsilon}),O(nk^2)\}$ ($\epsilon>0$ is a constant) time constant-factor approximations. For a planar graph $G$ and $k=bw(G)$, 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 $\min\{O(n\log^3n\log k),O(nk^2\log k)\}$ time. 
Qian-Ping Gu and Gengchun Xu
cs.DS