A Refined Kernel for $d$-Hitting Set

Yuxi Liu; Mingyu Xiao

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arXiv · 2506.24114

A Refined Kernel for $d$-Hitting Set

Yuxi Liu·Mingyu Xiao

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Abstract

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.

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BibTeX RIS

Yuxi Liu, Mingyu Xiao. 2025-06-30. A Refined Kernel for $d$-Hitting Set. https://arxiv.org/abs/2506.24114

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