A Near-Linear Element Kernel for \(d\)-Hitting Set

Abstract

In \(d\)-\textsc{Hitting Set}, the input consists of a finite universe \(U\), a family \(\mathcal S\) of subsets of \(U\) with size at most \(d\), and an integer \(k\). The task is to decide whether at most \(k\) elements of \(U\) can intersect every set in \(\mathcal S\). For every fixed \(d\geq3\), we give a one-sided randomized kernel with \(O(k\log^3k)\) elements and a deterministic kernel with \(O(k^2\log k)\) elements for \(d\)-\textsc{Hitting Set}. In the one-sided randomized kernel, every NO-instance is always mapped to a NO-instance, and a YES-instance is mapped to a YES-instance with constant probability. The previously known kernels for \(d\)-\textsc{Hitting Set} contain \(O(k^{d-1})\) elements and \(O(k^d)\) sets. It has been asked in the literature whether \(d\)-\textsc{Hitting Set} allows kernels with \(O(k^{d-1-\varepsilon})\) elements for some constant \(\varepsilon>0\). In this paper, we answer this question affirmatively by giving near-linear element-kernels through a re-encoding of the instance. On the other hand, our kernel may still contain \(k^{O(d)}\) sets and the parameter $k$ may grow polynomially.