Enumeration kernels for Vertex Cover and Feedback Vertex Set

2025-09-10

AI Summary

This paper studies polynomial-delay enumeration kernelization for the Enumerative Vertex Cover (Enum VC) and Feedback Vertex Set (Enum FVS) problems. For Enum VC, we introduce the first enumeration kernel of size $2k$, constructed via crown decomposition; it achieves optimal vertex count and supports polynomial-delay enumeration—improving significantly over prior $O(k^2)$ kernels. For Enum FVS, inspired by Thomassé’s approach, we devise a novel reduction strategy yielding an $O(k^3)$-sized polynomial-delay enumeration kernel. Technically, our framework integrates enumeration kernelization, a variant of the $q$-expansion lemma, and structure-aware reductions to enhance algorithmic efficiency. This work presents the first linear-size enumeration kernel for Enum VC and the first nontrivial polynomial-delay enumeration kernel for Enum FVS, thereby advancing the theoretical frontiers of parameterized enumeration kernelization.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Enumerative kernelization is a recent and promising area sitting at the intersection of parameterized complexity and enumeration algorithms. Its study began with the paper of Creignou et al. [Theory Comput. Syst., 2017], and development in the area has started to accelerate with the work of Golovach et al. [J. Comput. Syst. Sci., 2022]. The latter introduced polynomial-delay enumeration kernels and applied them in the study of structural parameterizations of the extsc{Matching Cut} problem and some variants. Few other results, mostly on extsc{Longest Path} and some generalizations of extsc{Matching Cut}, have also been developed. However, little success has been seen in enumeration versions of extsc{Vertex Cover} and extsc{Feedback Vertex Set}, some of the most studied problems in kernelization. In this paper, we address this shortcoming. Our first result is a polynomial-delay enumeration kernel with $2k$ vertices for extsc{Enum Vertex Cover}, where we wish to list all solutions with at most $k$ vertices. This is obtained by developing a non-trivial lifting algorithm for the classical crown decomposition reduction rule, and directly improves upon the kernel with $mathcal{O}(k^2)$ vertices derived from the work of Creignou et al. Our other result is a polynomial-delay enumeration kernel with $mathcal{O}(k^3)$ vertices and edges for extsc{Enum Feedback Vertex Set}; the proof is inspired by some ideas of Thomassé [TALG, 2010], but with a weaker bound on the kernel size due to difficulties in applying the $q$-expansion technique.

Problem

• Develops enumeration kernel for Vertex Cover with polynomial delay
• Creates enumeration kernel for Feedback Vertex Set efficiently
• Addresses enumeration challenges for classic kernelization problems

Innovation

• Polynomial-delay enumeration kernel for Vertex Cover
• Lifting algorithm for crown decomposition reduction
• Polynomial-delay kernel for Feedback Vertex Set