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

Abstract

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 $O(2^{13.6\sqrt{k}}\sqrt{k}n+n^{3})$.

This work was partially supported by Norges forskningsråd project 160778/V30, and partially by the Netherlands Organisation for Scientific Research NWO (project Treewidth and Combinatorial Optimisation).