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

Abstract

We give an algorithm which for an input planar graph $G$ of $n$ vertices and integer $k$, in $\min \{O(n{\log }^{3}n),O(n{k}^2)\}$ 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 $\min \{O(n^{1+\epsilon }),O(n{k}^2)\}$ ($\epsilon >0$ 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 $\min \{O(n{\log }^{3}n\log k),O(n{k}^2\log k)\}$ time.

arXiv commentThe mainly revision is the $O(n{k}^2)$ algorithm part (Section 4): added proofs for graphs with edge weights 1/2 and 1, and modified the proofs for finding the minimum separating cycles