Affiliation:
1. Technische Universität Berlin, Berlin, Germany
Abstract
Nowhere dense classes of graphs [21, 22] are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of
uniform quasi-wideness
, a concept originating in finite model theory, has proved to be particularly useful. Uniform quasi-wideness is used in many fpt-algorithms on nowhere dense classes. However, the existing constructions showing the equivalence of nowhere denseness and uniform quasi-wideness imply a non-elementary blow up in the parameter dependence of the fpt-algorithms, making them infeasible in practice. As a first main result of this article, we use tools from logic, in particular from a sub-field of model theory known as stability theory, to establish polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness. A powerful method in parameterized complexity theory is to compute a problem kernel in a pre-computation step, that is, to reduce the input instance in polynomial time to a sub-instance of size bounded in the parameter only (independently of the input graph size). Our new tools allow us to obtain for every fixed radius
r
∈ N a polynomial kernel for the distance-
r
dominating set problem on nowhere dense classes of graphs. This result is particularly interesting, as it implies that for every class
C
of graphs that is closed under taking subgraphs, the distance-
r
dominating set problem admits a kernel on
C
for every value of
r
if, and only if, it already admits a polynomial kernel for every value of
r
(under the standard assumption of parameterized complexity theory that FPT ≠ W[2]).
Funder
European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
3 articles.
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