On the VC-dimension of half-spaces with respect to convex sets

Nicolas Grelier, Saeed Gh Ilchi, Tillmann Miltzow, Shakhar Smorodinsky

Research output: Contribution to journalArticlepeer-review

Abstract

A family S of convex sets in the plane defines a hypergraph H = (S, Ɛ) with S as a vertex set and Ɛ as the set of hyperedges as follows. Every subfamily S' ⊂ S defines a hyperedge in E if and only if there exists a halfspace h that fully contains S', and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in ℝd, for d ≥ 3. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings. (Figure presented)

Original languageEnglish
Article number2
JournalDiscrete Mathematics and Theoretical Computer Science
Volume23
Issue number1
DOIs
StatePublished - 1 Jan 2021

Keywords

  • Convex sets
  • Epsilon nets
  • Halfplanes
  • VC-dimension

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)
  • Discrete Mathematics and Combinatorics

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