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- W4320341108 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 $kgeq 2$ 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." @default.
- W4320341108 created "2023-02-13" @default.
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- W4320341108 date "2022-08-30" @default.
- W4320341108 modified "2023-09-23" @default.
- W4320341108 title "On Sparse Hitting Sets: from Fair Vertex Cover to Highway Dimension" @default.
- W4320341108 doi "https://doi.org/10.48550/arxiv.2208.14132" @default.
- W4320341108 hasPublicationYear "2022" @default.
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