Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Branch Decompositions

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Divide-and-conquer strategy based on variations of the Lipton-Tarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour & Thomas, combined with new techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. Compared to divide-and-conquer algorithms, the main advantages of our method are a) it is a generic method which allows to attack broad classes of problems; b) the obtained algorithms provide a better worst case analysis. To exemplify our approach we show how to obtain an  \(O(2^{6.903\sqrt{n}}n^{3/2}+n^{3})\) time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time \(O(2^{10.8224\sqrt{n}}n^{3/2}+n^{3})\). Our approach can be used to design parameterized algorithms as well. For example we introduce the first \(2^{O\sqrt{k}}k^{O(1)}.n^{O(1)}\) time algorithm for parameterized Planar k –cycle by showing that for a given k we can decide if a planar graph on n vertices has a cycle of length ≥ k in time \(O(2^{13.6\sqrt{k}}\sqrt{k}n+n^{3})\).