Affiliation:
1. MIT, Cambridge, Massachusetts
2. University of Bergen, Bergen, Norway
3. National and Capodistrian University of Athens, Athens, Greece
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.
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Cited by
224 articles.
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