Subexponential parameterized algorithms on bounded-genus graphs and <i>H</i> -minor-free graphs

Abstract

We introduce a new framework for designing fixed-parameter algorithms with subexponential running time---2 O(√k) n O(1) . Our results apply to a broad family of graph problems, called bidimensional problems , which includes many domination and problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, disk dimension, and many others restricted to bounded-genus graphs (phrased as bipartite-graph problem ). Furthermore, it is fairly straightforward to prove that a problem is bidimensional. In particular, our framework includes, as special cases, all previously known problems to have such subexponential algorithms. Previously, these algorithms applied to planar graphs, single-crossing-minor-free graphs, and/or map graphs; we extend these results to apply to bounded-genus graphs as well. In a parallel development of combinatorial results, we establish an upper bound on the treewidth (or branchwidth) of a bounded-genus graph that excludes some planar graph H as a minor. This bound depends linearly on the size |V(H)| of the excluded graph H and the genus g(G) of the graph G , and applies and extends the graph-minors work of Robertson and Seymour.Building on these results, we develop subexponential fixed-parameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. In particular, this general category of graphs includes planar graphs, bounded-genus graphs, single-crossing-minor-free graphs, and any class of graphs that is closed under taking minors. Specifically, the running time is 2 O(√k) n h , where h is a constant depending only on H , which is polynomial for k = O (log 2 n ). We introduce a general approach for developing algorithms on H -minor-free graphs, based on structural results about H -minor-free graphs at the heart of Robertson and Seymour's graph-minors work. We believe this approach opens the way to further development on problems in H -minor-free graphs.