A Geometric Approach to Parameterized Algorithms for Domination Problems on Planar Graphs

This paper revisits design and analysis techniques for fixed parameter algorithms for Planar Dominating Set and other problems on planar structures. As our main result, we use new geometric arguments concerning treewidth-based algorithms to show that determining whether a planar graph G has a dominating set of size k can be solved in \(O(2^{16.4715 \sqrt{k}}+ n^3)\) steps. This result improves on the best known treewidth-based algorithm by Kanj and Perkovic that runs in time \(O(2^{27\sqrt{k}}n)\). Our main result nearly matches the new branchwidth-based algorithm for Planar Dominating Set by Fomin and Thilikos that runs in time \(O(2^{15.13 \sqrt{k}}k +n^3)\). Algorithms for other problems on planar structures are explored. In particular, we show that Planar Red/Blue Dominating Set can be solved in time \(O(2^{24.551 \sqrt{k}}n)\). This leads to the main results, namely, that faster parameterized algorithms can be obtained for a variety of problems that can be described by planar boolean formulae. This gives the best-known parameterized algorithms for Planar Vertex Cover, Planar Edge Dominating Set, and Face Cover.