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

• Published in: Discrete mathematics and theoretical computer science Vol. 23, no. 3; no. Combinatorics
• Main Authors: Grelier, Nicolas, Ilchi, Saeed Gh, Miltzow, Tillmann, Smorodinsky, Shakhar
• Format: Journal Article
• Language: English
• Published: Discrete Mathematics & Theoretical Computer Science 19.08.2021
• Subjects: computer science - computational geometry; computer science - data structures and algorithms; computer science - discrete mathematics; mathematics - combinatorics
• ISSN: 1365-8050; 1365-8050
• DOI: 10.46298/dmtcs.6631

A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H 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 R^d , for d > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always 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.