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(26.903√nn3/2 +n3) 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√nn3/2 +n3). Our approach can be used to design parameterized algorithms as well. For example we introduce the first 2O(√k)kO(1) · nO(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(213.6√k√kn + n3). © Springer-Verlag Berlin Heidelberg 2005.