Abstract. Kernelization is a strong and widely-applied technique in the design of fixed-parameter algorithms. In a nutshell, a kernelization algorithm is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernelization algorithm is called a polynomial kernel if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper, we give evidence that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. P vs. NP), and evolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which might be of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine which allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These include important FPT problems such as k-Path, k-Minor Order Test, k-Planar Graph Subgraph Test, and several others. We also show that a polynomial kernel for parameterized problems fulfilling a slightly different criteria implies a distillation algorithm for all coNP-complete problems. Example of such problems are kPathwidth, k-Treewidth, and k-Cutwidth. Moreover, our framework also allows us to establish kernelization lower bounds for optimization problems (e.g. Minimum Dominating Set) parameterized by treewidth, cliquewidth, max. degree, and so forth. In the last part of the paper, we study subexponential kernelization algorithms, and look into these types of kernelizations from a more structural point of view. Our results are the first negative results concerning polynomial kernels for natural FPT problems. Apart from these, the only previously known techniques for showing lower bounds were limited to linear lower bounds. We believe our results provide a glimpse of what makes polynomial kernelization intrinsically hard in certain type of problems, and also offer an accessible platform for showing similar results.
Very recently, Fellows et al. REF propose a new method which allows us to show that many problems do not have polynomial size kernels under reasonable complexity-theoretic assumptions.
On problems without polynomial kernels
Lect. Notes Comput. Sci
Mathematics Computer Science