Approximate Turing Kernelization for Problems Parameterized by Treewidth
Abstract
We extend the notion of lossy kernelization, introduced by Lokshtanov et al. [STOC 2017], to approximate Turing kernelization. An α-approximate Turing kernel for a parameterized optimization problem is a polynomial-time algorithm that, when given access to an oracle that outputs c-approximate solutions in O(1) time, obtains an (α · c)-approximate solution to the considered problem, using calls to the oracle of size at most f(k) for some function f that only depends on the parameter. Using this definition, we show that Independent Set parameterized by treewidth has a (1+)-approximate Turing kernel with O(2) vertices, answering an open question posed by Lokshtanov et al. [STOC 2017]. Furthermore, we give (1+)-approximate Turing kernels for the following graph problems parameterized by treewidth: Vertex Cover, Edge Clique Cover, Edge-Disjoint Triangle Packing and Connected Vertex Cover. We generalize the result for Independent Set and Vertex Cover, by showing that all graph problems that we will call "friendly" admit (1+)-approximate Turing kernels of polynomial size when parameterized by treewidth. We use this to obtain approximate Turing kernels for Vertex-Disjoint H-packing for connected graphs H, Clique Cover, Feedback Vertex Set and Edge Dominating Set.