A Polynomial Kernel for Deletion to the Scattered Class of Cliques and Trees

2024-09-21

International Symposium on Algorithms and Computation

AI Summary

This paper studies the *scattered deletion* problem in graph modification: given a graph $G$ and an integer $k$, determine whether at most $k$ vertices can be deleted so that each connected component of the resulting graph is either a clique or a tree. This is the first nontrivial polynomial kernel for this dispersed target class. We develop a unified framework combining modular decomposition, structural graph analysis, and redundancy-based vertex compression to construct a deterministic polynomial kernel of size $O(k^5)$. Our result fills a fundamental gap in kernelization research for scattered deletion problems, achieves significant input size reduction, and provides a robust preprocessing foundation for designing efficient fixed-parameter tractable (FPT) algorithms.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Kernel Methods
• Search and Optimization: Combinatorial Optimization

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: Normalization, clustering, classification, and summarization of Web text

Abstract

The class of graph deletion problems has been extensively studied in theoretical computer science, particularly in the field of parameterized complexity. Recently, a new notion of graph deletion problems was introduced, called deletion to scattered graph classes, where after deletion, each connected component of the graph should belong to at least one of the given graph classes. While fixed-parameter algorithms were given for a wide variety of problems, little progress has been made on the kernelization complexity of any of them. In this paper, we present the first non-trivial polynomial kernel for one such deletion problem, where, after deletion, each connected component should be a clique or a tree - that is, as densest as possible or as sparsest as possible (while being connected). We develop a kernel consisting of O(k^5) vertices for this problem.

Problem

• Develops polynomial kernel for scattered graph deletion
• Ensures components become cliques or trees
• Addresses kernelization complexity with O(k^5) vertices

Innovation

• Polynomial kernel for deletion problems
• Components become cliques or trees
• Kernel with O(k^5) vertices size