A Refined Kernel for $d$-Hitting Set

TL;DR

This paper improves the kernel size for the $d$-Hitting Set problem using linear programming to refine crown decompositions, reducing the vertex count from $(2d - 1)k^{d - 1} + k$ to $(2d - 2)k^{d - 1} + k$.

Contribution

The authors introduce a linear programming-based method to refine existing kernelization bounds for the $d$-Hitting Set problem.

Findings

Kernel size improved from $(2d - 1)k^{d - 1} + k$ to $(2d - 2)k^{d - 1} + k$

Linear programming techniques enable more efficient crown decompositions

Refinement has potential applications in related parameterized problems

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.