A More Versatile Model for Enumerative Kernelization: A Case Study for Vertex Cover

Abstract

Enumerative kernelization is a relatively recent and promising area sitting at the intersection of parameterized complexity and enumeration algorithms, with two main models being proposed. The first, known as enum-kernels and due to Creignou et al. [Theory Comput. Syst., 2017], was too permissive, leading to constant-sized kernels for every problem solvable with FPT-delay. To remedy this, Golovach et al. [J. Comput. Syst. Sci., 2022] 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 that the area has enjoyed so far. 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 increased versatility and the expressive power of our framework on the Enum Vertex Cover problem, where we want to list all vertex covers of size at most k of a given graph. In particular, we manage to generalize the kernelization dichotomy by Bougeret et al. [SIAM J. Discrete Math., 2022] about the existence of polynomial kernels for Vertex Cover parameterized by the vertex deletion distance to a minor-closed graph class, as well as the solution size and feedback vertex number parameterizations. The second one, in particular, is significantly simpler than the kernel designed by Bougeret et al. [IPEC, 2025], 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. [Algorithmica, 2019] for the vertex-deletion distance to c-treedepth.