2016/08/19 by Stephan Kreutzer, Kreutzer, Stephan, Roman Rabinovich +3
Computer Science · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Machine Learning and Algorithms
paper · pdf · doi:10.48550/arxiv.1608.05637
Nowhere dense classes of graphs 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 paper, we use tools from logic, in particular from a subfield 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 value of r 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 which is closed under subgraphs, the distance-r dominating set problem admits a kernel on C for every value of r if, and only if, it admits a polynomial kernel for every value of r (under the standard assumption of parameterized complexity theory that FPT ≠ W[2]).