On the VC-dimension of convex sets and half-spaces

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

Research output: Working paper/PreprintPreprint

Abstract

A family S of convex sets in the plane defines a hypergraph H = (S, ε with S as a vertex set and as the set of hyperedges as follows. Every subfamily S' ⊂ S defines a hyperedge in ε  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.
Original languageEnglish
PublisherarXiv:1612.08154 [cs]
StatePublished - 2 Jul 2019

Keywords

  • cs.CG
  • cs.DM
  • cs.DS
  • math.CO

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