Johannes Blum, Y. Disser, A. Feldmann, Siddharth Gupta, Anna Zych-Pawlewicz
{"title":"关于稀疏命中集:从公平顶点覆盖到高速公路维度","authors":"Johannes Blum, Y. Disser, A. Feldmann, Siddharth Gupta, Anna Zych-Pawlewicz","doi":"10.48550/arXiv.2208.14132","DOIUrl":null,"url":null,"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.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"29 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"On Sparse Hitting Sets: from Fair Vertex Cover to Highway Dimension\",\"authors\":\"Johannes Blum, Y. Disser, A. Feldmann, Siddharth Gupta, Anna Zych-Pawlewicz\",\"doi\":\"10.48550/arXiv.2208.14132\",\"DOIUrl\":null,\"url\":null,\"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. 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On Sparse Hitting Sets: from Fair Vertex Cover to Highway Dimension
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.