A more versatile model for enumerative kernelization: a case study for Vertex Cover

2026-04-25

AI Summary

Existing enumeration kernelization models are either too permissive to be practical or overly restrictive, thereby limiting algorithmic design. This work proposes a new polynomial-delay (PD) kernelization model that retains rigor while permitting the exclusion of infeasible solutions in compressed instances. We develop a general framework to transform decision kernels into PD kernels of comparable size, overcoming key limitations of traditional enumeration kernelization theory. Our approach provides, for the first time, a unified and simplified treatment of enumeration kernelizations for Vertex Cover under various parameters—including distance to small closed graph classes, solution size, and feedback vertex set number—significantly improving upon existing lifting algorithms.

Technology Category

• Machine Learning: Kernel Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Enumerative kernelization is a recent promising at the intersection of parameterized complexity and enumeration algorithms, with two proposed models. The first, known as enum-kernels and due to Creignou et al., was too permissive, leading to constant-sized kernels for every problem solvable with FPT-delay. To remedy this, Golovach et al. proposed the polynomial-delay enumeration kernelization model that, while addressing the shortcoming of the previous one, appears to be too strict, which we believe is a central reason for the slow development of the area. In this paper, we propose a new model for enumeration kernels, which we have called polynomial-delay (PD) kernels. It is more flexible than Golovach et al.'s kernels while still preserving their qualities; informally, it allows us to ignore ``bad'' solutions of the compressed instance when producing the solution set of the input instance, but still requires that the ``good'' solutions are lifted with polynomial-delay. After discussing the main properties of our model, we design a generic framework for vertex-subset problems to adapt decision kernels into PD kernels of the same size. We showcase our model's versatility and the framework's expressive power on the \textsc{Enum Vertex Cover} problem, where we want to list all vertex covers of size at most $k$ of a given graph. We generalize the kernelization dichotomy by Bougeret et al. about the existence of polynomial kernels for \textsc{Vertex Cover} parameterized by the vertex deletion distance to a minor-closed graph class, as well as by the solution size or feedback vertex number. The second one, in particular, is significantly simpler than the known kernel, requiring only a few lines for its lifting algorithm. Beyond our framework, we also show how to generalize to the enumeration setting the kernel of Bougeret et al. for the vertex-deletion distance to $c$-treedepth.

Problem

• enumerative kernelization
• parameterized complexity
• enumeration algorithms
• vertex cover
• polynomial-delay

Innovation

• enumerative kernelization
• polynomial-delay kernels
• Vertex Cover enumeration
• parameterized complexity
• kernelization framework