Constant-Factor Approximations of Branch-Decomposition and Largest Grid Minor of Planar Graphs in O(n1 + ε) Time

• Authors: Qian-Ping Gu;Hisao Tamaki
• Venue: ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
• Year: 2009

We give constant-factor approximation algorithms for computing the optimal branch-decompositions and largest grid minors of planar graphs. For a planar graph G with n vertices, let ${\mathop{\rm bw}}(G)$ be the branchwidth of G and ${\mathop{\rm gm}}(G)$ the largest integer g such that G has a g脳g grid as a minor. Let c 驴 1 be a fixed integer and 驴,β be arbitrary constants satisfying 驴 c + 1.5 and β 2c + 1.5. We give an algorithm which constructs in $O(n^{1+\frac{1}{c}}\log n)$ time a branch-decomposition of G with width at most $\alpha\,{\mathop{\rm bw}}(G)$. We also give an algorithm which constructs a g脳g grid minor of G with $g\geq \frac{{\mathop{\rm gm}}(G)}{\beta}$ in $O(n^{1+\frac{1}{c}}\log n)$ time. The constants hidden in the Big-Oh notations are proportional to $\frac{c}{\alpha-(c+1.5)}$ and $\frac{c}{\beta-(2c+1.5)}$, respectively.