On Sparse Hitting Sets: from Fair Vertex Cover to Highway Dimension

Johannes Blum, Y. Disser, A. Feldmann, Siddharth Gupta, Anna Zych-Pawlewicz
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引用次数: 2

Abstract

We consider the Sparse Hitting Set (Sparse-HS) problem, where we are given a set system $(V,\mathcal{F},\mathcal{B})$ with two families $\mathcal{F},\mathcal{B}$ of subsets of $V$. The task is to find a hitting set for $\mathcal{F}$ that minimizes the maximum number of elements in any of the sets of $\mathcal{B}$. Our focus is on determining the complexity of some special cases of Sparse-HS with respect to the sparseness $k$, which is the optimum number of hitting set elements in any set of $\mathcal{B}$. For the Sparse Vertex Cover (Sparse-VC) problem, $V$ is given by the vertex set of a graph, and $\mathcal{F}$ is its edge set. We prove NP-hardness for sparseness $k\geq 2$ and polynomial time solvability for $k=1$. We also provide a polynomial-time $2$-approximation for any $k$. A special case of Sparse-VC is Fair Vertex Cover (Fair-VC), where the family $\mathcal{B}$ is given by vertex neighbourhoods. For this problem we prove NP-hardness for constant $k$ and provide a polynomial-time $(2-\frac{1}{k})$-approximation. This is better than any approximation possible for Sparse-VC or Vertex Cover (under UGC). We then consider two problems derived from Sparse-HS related to the highway dimension, a graph parameter modelling transportation networks. Most algorithms for graphs of low highway dimension compute solutions to the $r$-Shortest Path Cover ($r$-SPC) problem, where $r>0$, $\mathcal{F}$ contains all shortest paths of length between $r$ and $2r$, and $\mathcal{B}$ contains all balls of radius $2r$. There is an XP algorithm that computes solutions to $r$-SPC of sparseness at most $h$ if the input graph has highway dimension $h$, but the existence if an FPT algorithm was open. We prove that $r$-SPC and also the related $r$-Highway Dimension ($r$-HD) problem are both W[1]-hard. Furthermore, we prove that $r$-SPC admits a polynomial-time $O(\log n)$-approximation.
关于稀疏命中集:从公平顶点覆盖到高速公路维度
我们考虑稀疏命中集(Sparse- hs)问题,其中我们给定一个集系统$(V,\mathcal{F},\mathcal{B})$,其中$V$的两个子集族$\mathcal{F},\mathcal{B}$。任务是为$\mathcal{F}$找到一个命中集,该命中集使$\mathcal{B}$的任何集合中的最大元素数量最小化。我们的重点是确定关于稀疏性$k$的一些特殊情况下的Sparse-HS的复杂性,这是在$\mathcal{B}$的任何集合中命中集合元素的最佳数量。对于稀疏顶点覆盖(Sparse- vc)问题,$V$由图的顶点集给出,$\mathcal{F}$为图的边集。我们证明了稀疏性的np硬度$k\geq 2$和$k=1$的多项式时间可解性。我们还为任意$k$提供了一个多项式时间$2$ -近似。稀疏vc的一个特例是公平顶点覆盖(Fair- vc),其中族$\mathcal{B}$是由顶点邻域给出的。对于这个问题,我们证明了常数$k$的np -硬度,并提供了一个多项式-时间$(2-\frac{1}{k})$ -近似。这比Sparse-VC或Vertex Cover(在UGC下)的任何近似都要好。然后,我们考虑了两个由稀疏- hs衍生的与公路维度相关的问题,这是一个建模交通网络的图参数。大多数用于低公路维图的算法计算$r$ -最短路径覆盖($r$ -SPC)问题的解决方案,其中$r>0$, $\mathcal{F}$包含长度在$r$和$2r$之间的所有最短路径,$\mathcal{B}$包含半径为$2r$的所有球。有一种XP算法,在输入图具有高速公路维数$h$的情况下,最多计算稀疏度的$r$ -SPC的解$h$,但如果FPT算法是开放的,则存在。我们证明了$r$ -SPC和相关的$r$ -公路维数($r$ -HD)问题都是W[1]-难的。进一步证明$r$ -SPC允许多项式时间$O(\log n)$ -近似。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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