Abstract
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 \(O(n\log ^4n)\) 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 \({\mathrm{bw}}(G)>k\), \({\mathrm{bw}}(G)\) 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 \(\min \{O(n^{1+\epsilon }),O(nk^3)\}\) (\(\epsilon >0\) is a constant) time constant-factor approximations. For a planar graph \(G\) and \(k={\mathrm{bw}}(G)\), 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 \(O(n\log ^4n\log k)\) time.
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Gu, QP., Xu, G. (2014). Near-Linear Time Constant-Factor Approximation Algorithm for Branch-Decomposition of Planar Graphs. In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_20
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