A more versatile model for enumerative kernelization: a case study for Vertex Cover
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| Format: | Preprint |
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2026
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| author | Bougeret, Marin Gomes, Guilherme C. M. Sau, Ignasi |
| author_facet | Bougeret, Marin Gomes, Guilherme C. M. Sau, Ignasi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_23419 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A more versatile model for enumerative kernelization: a case study for Vertex Cover Bougeret, Marin Gomes, Guilherme C. M. Sau, Ignasi Data Structures and Algorithms 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. |
| title | A more versatile model for enumerative kernelization: a case study for Vertex Cover |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.23419 |