FPT-space Graph Kernelizations

Let $n$ be the size of a parametrized problem and $k$ the parameter. We present polynomial-time kernelizations for Cluster Editing/Deletion, Path Contractions and Feedback Vertex Set that run with $O(\mathrm{poly}(k) \log n)$ bits and compute a kernel of size polynomial in $k$. By first executing the new kernelizations and subsequently the best known polynomial-time kernelizations for the problem under consideration, we obtain the best known kernels in polynomial time with $O(\mathrm{poly}(k) \log n)$ bits. Our kernelization for Feedback Vertex Set computes in a first step an approximated solution, which can be used to build a simple algorithm for undirected $s$-$t$-connectivity (USTCON) that runs in polynomial time and with $O(\mathrm{poly}(k) \log n)$ bits.