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

Abstract

In \(d\)-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 \(\mathcal S\). Previously known kernels contain \(O(k^{d-1})\) elements and \(O(k^d)\) sets. Whether the number of elements can be reduced to \(O(k^{d-1-\varepsilon})\) for some constant \(\varepsilon>0\) is a longstanding open problem in kernelization. In this paper, we answer this question affirmatively by giving a near-linear randomized element-kernel and a subquadratic deterministic element-kernel through a re-encoding of the instance. For every fixed \(d\geq3\), we give a one-sided randomized kernel with \(O_d(k\log^3k)\) elements and a deterministic kernel with \(O_d(k^{7/4}\log^{3/2}k)\) elements. Our kernels may still contain \(k^{O(d)}\) sets and the parameter \(k\) may grow polynomially.