Lossy Kernels for Connected Dominating Set on Sparse Graphs

For alpha > 1, an alpha-approximate (bi-)kernel for a problem Q is a polynomial-time algorithm that takes as input an instance (I, k) of Q and outputs an instance (I', k') (of a problem Q') of size bounded by a function of k such that, for every c >= 1, a alpha-approximate solution for the new instance can be turned into a (c . alpha)-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov et al. 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 subset of coNP/poly. We complement our results by the following conditional lower bound. We show that if a class C is somewhere dense and closed under taking subgraphs, then for some value of r is an element of N there cannot exist an alpha-approximate bi-kernel for the (CONNECTED) DISTANCE -r DOMINATING SET problem on C for any alpha > 1 (assuming the Gap Exponential Time Hypothesis).