A Near-Linear Element Kernel for $d$-Hitting Set

Zimo Sheng; Mingyu Xiao

Home/Archive/arXiv

arXiv · 2609.06512

A Near-Linear Element Kernel for $d$-Hitting Set

Zimo Sheng·Mingyu Xiao

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.DS cs.CC

Keep this discovery

BibTeX RIS

Zimo Sheng, Mingyu Xiao. 2026-09-06. A Near-Linear Element Kernel for $d$-Hitting Set. https://arxiv.org/abs/2609.06512

Cite the original work for its findings. Save a collection to share your selection of sources.

Discover connections

cs.CC subject cs.DS subject

Connections use source metadata and explicit phrase matches, not verified experimental comparisons.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗