Boundaried Kernelization

2025-04-25

AI Summary

This work addresses the limitation of traditional polynomial kernelization—its reliance on global graph structure, which hinders adaptability to local preprocessing requirements—by introducing “boundary kernelization,” a novel kernelization paradigm. The model systematically investigates polynomial kernelization on boundary graphs, overcoming the exponential-size barrier inherent in prior protrusion-replacement approaches and establishing polynomial-size kernels on broader graph classes. It integrates boundary graph theory, protrusion analysis, tree-depth distance metrics, and meta-kernelization principles to construct a provably sound local replacement framework. Key contributions include: (i) the first polynomial-size boundary kernels for several classical graph problems; (ii) refutations of the existence of polynomial kernels for certain problems under standard complexity assumptions; and (iii) a significant improvement in the kernel size for Vertex Cover parameterized by deletion distance to treedepth.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Proposes boundaried kernelization for local preprocessing
• Studies polynomial preprocessing for boundaried graphs
• Improves kernelization for Vertex Cover parameterized

Innovation

• Introduces boundaried kernelization for local preprocessing
• Focuses on polynomial bounds in general settings
• Applies to Vertex Cover with improved kernelization