Lossy Kernels for Connected Dominating Set on Sparse Graphs

In standard kernelization algorithms, the usual goal is to reduce, in polynomial time, an instance (I, k) of a parameterized problem to an equivalent instance (I ′, k′) of size bounded by a function in k. One of the central problems in this area, whose investigation has led to the development of many kernelization techniques, is the Dominating Set problem. Given a graph G and k ∈ N, Dominating Set asks for a subset D of k vertices such that every vertex in G is either in D or has a neighbor in D. It is well known that Dominating Set is W[2]-hard when parameterized by k. But it admits a linear kernel on graphs of bounded expansion and a polynomial kernel on Kd,dfree graphs, for a fixed constant d. In contrast, the closely related Connected Dominating Set problem (where G[D] is required to be connected) is known not to admit such kernels unless NP ⊆ coNP/poly. We show that even though the kernelization complexity of Dominating Set and Connected Dominating Set diverges on sparse graphs this divergence is not as extreme as kernelization lower bounds suggest. To do so, we study the Connected Dominating Set problem under the recently introduced framework of lossy kernelization. In this framework, for α > 1, an α-approximate bikernel (kernel) is a polynomial-time algorithm that takes as input an instance (I, k) and outputs an instance (I ′, k′) of a problem (the same problem) such that, for every c > 1, a c-approximate solution for the new instance can be turned into a cα-approximate solution of the original instance in polynomial time. Moreover, the size of (I ′, k′) is bounded by a function of k and α. We show that Connected Dominating Set admits an α-approximate bikernel on graphs of bounded expansion and an α-approximate kernel on Kd,d-free graphs, for every α > 1. ForKd,d-free graphs we obtain instances of size kO( d2 α−1 ) while for bounded expansion graphs we obtain instances of size O(f(α)k) (i.e, linear in k), where f(α) is a computable function depending only on α. 1998 ACM Subject Classification G.2.2 Graph Algorithms, I.1.2 Analysis of Algorithms