LOSSY KERNELS FOR CONNECTED DOMINATING SET ON SPARSE GRAPHS.

For \alpha > 1, an \alpha -approximate (bi)kernel is a polynomial-time algorithm that takes as input an instance (I, k) of a problem \scrQ and outputs an instance (I\prime, k\prime) (of a problem \scrQ \prime) of size bounded by a function of k such that, for every c \geq 1, a c-approximate solution for the new instance can be turned into a (c \cdot \alpha)-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov and co-authors. We study Connected Dominating Set (and its distance-r variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every \alpha > 1, Connected Dominating Set admits a polynomial-size \alpha -approximate (bi)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on 2-degenerate graphs and graphs of bounded expansion, unless NP \subseteq coNP/poly. We complement our results by the following conditional lower bound. We show that if a class \scrC is somewhere dense and closed under taking subgraphs, then for some value of r \in N there cannot exist an \alpha -approximate bi-kernel for the (Connected) Distance-r Dominating Set problem on \scrC for any \alpha > 1 (assuming FPT \not = W[1]). [ABSTRACT FROM AUTHOR]