Boundaried Kernelization
Leonid Antipov, Stefan Kratsch
TL;DR
Boundaried kernelization presents a local preprocessing paradigm that uses boundaried graphs and gluing to capture local structure and yield global equivalence guarantees. The authors develop formal definitions, connect boundaried kernels to standard kernels, and derive both upper and lower bounds across a spectrum of graph problems, highlighting when polynomial boundaried kernels exist or are impossible. They show concrete polynomial bounds for problems like Vertex Cover (vc), Vertex Cover with a forest/feedback vertex modulator (vc[fvs], fvs), and several Long Cycle/Path and Hamiltonian variants, while proving strong lower bounds for Cluster Editing, Maximum Cut, Tree Deletion Set, Long Cycle/Path variants, and Dominating Set with respect to boundaried kernelization. A notable result is an improved polynomial boundaried kernel for VC with a constant treedepth modulator, reducing known kernel sizes from $k^{2^{igO(d^2)}}$ to $k^{2^{d-1}}$, underscoring the potential of local structure in obtaining tighter preprocessing. Overall, the work connects local, boundary-based reductions to global kernelization theory and lays groundwork for extending boundaried kernelization to broader finite structures and problems.
Abstract
The notion of a (polynomial) kernelization from parameterized complexity is a well-studied model for efficient preprocessing for hard computational problems. By now, it is quite well understood which parameterized problems do or (conditionally) do not admit a polynomial kernelization. Unfortunately, polynomial kernelizations seem to require strong restrictions on the global structure of inputs. To avoid this restriction, we propose a model for efficient local preprocessing that is aimed at local structure in inputs. Our notion, dubbed boundaried kernelization, is inspired by protrusions and protrusion replacement, which are tools in meta-kernelization [Bodlaender et al. J'ACM 2016]. Unlike previous work, we study the preprocessing of suitable boundaried graphs in their own right, in significantly more general settings, and aiming for polynomial rather than exponential bounds. We establish polynomial boundaried kernelizations for a number of problems, while unconditionally ruling out such results for others. We also show that boundaried kernelization can be a tool for regular kernelization by using it to obtain an improved kernelization for Vertex Cover parameterized by the vertex-deletion distance to a graph of bounded treedepth.