Enumeration kernels for Vertex Cover and Feedback Vertex Set
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914031339569152 |
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| author | Bougeret, Marin Gomes, Guilherme C. M. Santos, Vinicius F. dos Sau, Ignasi |
| author_facet | Bougeret, Marin Gomes, Guilherme C. M. Santos, Vinicius F. dos Sau, Ignasi |
| contents | 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 \textsc{Matching Cut} problem and some variants. Few other results, mostly on \textsc{Longest Path} and some generalizations of \textsc{Matching Cut}, have also been developed. However, little success has been seen in enumeration versions of \textsc{Vertex Cover} and \textsc{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 \textsc{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 \textsc{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08475 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumeration kernels for Vertex Cover and Feedback Vertex Set Bougeret, Marin Gomes, Guilherme C. M. Santos, Vinicius F. dos Sau, Ignasi Data Structures and Algorithms 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 \textsc{Matching Cut} problem and some variants. Few other results, mostly on \textsc{Longest Path} and some generalizations of \textsc{Matching Cut}, have also been developed. However, little success has been seen in enumeration versions of \textsc{Vertex Cover} and \textsc{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 \textsc{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 \textsc{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. |
| title | Enumeration kernels for Vertex Cover and Feedback Vertex Set |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2509.08475 |