A Refined Kernel for $d$-Hitting Set
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908428361793536 |
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| author | Liu, Yuxi Xiao, Mingyu |
| author_facet | Liu, Yuxi Xiao, Mingyu |
| contents | The $d$-Hitting Set problem is a fundamental problem in parameterized complexity, which asks whether a given hypergraph contains a vertex subset $S$ of size at most $k$ that intersects every hyperedge (i.e., $S \cap e \neq \emptyset$ for each hyperedge $e$). The best known kernel for this problem, established by Abu-Khzam [1], has $(2d - 1)k^{d - 1} + k$ vertices. This result has been very widely used in the literature as many problems can be modeled as a special $d$-Hitting Set problem. In this work, we present a refinement to this result by employing linear programming techniques to construct crown decompositions in hypergraphs. This approach yields a slight but notable improvement, reducing the size to $(2d - 2)k^{d - 1} + k$ vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_24114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Refined Kernel for $d$-Hitting Set Liu, Yuxi Xiao, Mingyu Data Structures and Algorithms The $d$-Hitting Set problem is a fundamental problem in parameterized complexity, which asks whether a given hypergraph contains a vertex subset $S$ of size at most $k$ that intersects every hyperedge (i.e., $S \cap e \neq \emptyset$ for each hyperedge $e$). The best known kernel for this problem, established by Abu-Khzam [1], has $(2d - 1)k^{d - 1} + k$ vertices. This result has been very widely used in the literature as many problems can be modeled as a special $d$-Hitting Set problem. In this work, we present a refinement to this result by employing linear programming techniques to construct crown decompositions in hypergraphs. This approach yields a slight but notable improvement, reducing the size to $(2d - 2)k^{d - 1} + k$ vertices. |
| title | A Refined Kernel for $d$-Hitting Set |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2506.24114 |