Hitting Set for hypergraphs of low VC-dimension
Karl Bringmann, L\'aszl\'o Kozma, Shay Moran, and N.S. Narayanaswamy

TL;DR
This paper investigates the complexity of the Hitting Set problem in hypergraphs with low VC-dimension, revealing hardness results even at VC-dimension 2 and identifying a threshold for polynomial-time solvability.
Contribution
It characterizes the parameterized complexity of Hitting Set for low VC-dimension hypergraphs, proving W[1]-hardness at VC-dimension 2 and establishing a polynomial-time solvable class below a certain complexity threshold.
Findings
Hitting Set is W[1]-hard for VC-dimension 2 hypergraphs.
Hitting Set is polynomial-time solvable for VC-dimension 1.
A fine-grained measure identifies a sharp complexity threshold.
Abstract
We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small. VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a geometric or graph-theoretical flavor are known to have low VC-dimension. In set systems of bounded VC-dimension, Hitting Set is known to admit efficient and almost optimal approximation algorithms (Br\"onnimann and Goodrich, 1995; Even, Rawitz, and Shahar, 2005; Agarwal and Pan, 2014). In contrast to these approximation-results, a low VC-dimension does not necessarily imply tractability in the parameterized sense. In fact, we show that Hitting Set is W[1]-hard already on inputs with VC-dimension…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Advanced Graph Theory Research · Machine Learning and Algorithms
