# 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,ℱ,ℬ) with two families ℱ,ℬ of subsets of V. The task is to find a hitting set for ℱ that minimizes the maximum number of elements in any of the sets of 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 ℬ. For the Sparse Vertex Cover (Sparse-VC) problem, V is given by the vertex set of a graph, and ℱ is its edge set. We prove NP-hardness for sparseness k≥ 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 ℬ is given by vertex neighbourhoods. For this problem we prove NP-hardness for constant k and provide a polynomial-time (2-1/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, ℱ contains all shortest paths of length between r and 2r, and 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.

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