Matchings vs hitting sets among half-spaces in low dimensional euclidean spaces

Let $\mathcal{F}$ be any collection of linearly separable sets of a set $P$ of $n$ points either in $\mathbb{R}^2$, or in $\mathbb{R}^3$. We show that for every natural number $k$ either one can find $k$ pairwise disjoint sets in $\mathcal{F}$, or there are $O(k)$ points in $P$ that together hit all sets in $\mathcal{F}$. The proof is based on showing a similar result for families $\mathcal{F}$ of sets separable by pseudo-discs in $\mathbb{R}^2$. We complement these statements by showing that analogous result fails to hold for collections of linearly separable sets in $\mathbb{R}^4$ and higher dimensional euclidean spaces.

💡 Research Summary

The paper investigates the relationship between the size of a maximum matching (ν) and the size of a minimum hitting set (τ) in hypergraphs that arise from geometric range spaces defined by half‑spaces (or, equivalently, pseudo‑discs) in low‑dimensional Euclidean spaces. The authors focus on hypergraphs H(P,F) where P is a finite set of points and F is a family of ranges; edges correspond to the subsets P∩F. The central parameter is the affine sign‑rank, which equals the smallest dimension d in which the vertices can be embedded as points and the edges as half‑spaces so that incidence is preserved.

Main Positive Results (dimensions 2 and 3).

• Theorem 1 (2‑D pseudo‑discs). For any point set P⊂ℝ² and any family F of pseudo‑discs, and for any integer k≥1, either H(P,F) contains k pairwise disjoint edges, or there exists a set of O(k) points that hits every edge. Consequently τ(H)=Θ(ν(H)).
• Theorem 2 provides the key combinatorial tool: there exists an edge e such that the maximum matching among edges intersecting e has size at most 156. By repeatedly removing such an edge and all intersecting edges, one either builds a large matching or obtains a decomposition into O(k) sub‑hypergraphs each of bounded matching number. Each sub‑hypergraph can then be pierced by a constant number of points using a (p,q)‑theorem (Alon–Kleitman).
• Theorem 3 (3‑D half‑spaces). The same dichotomy holds for point sets in ℝ³ and families of half‑spaces.
• Theorem 4 is the 3‑D analogue of Theorem 2, again guaranteeing a “small” edge with at most 156 intersecting edges in any maximum matching.

The proofs rely on planar graph arguments (Hanani‑Tutte theorem) applied to an auxiliary graph whose vertices represent edges of a maximum matching and whose edges encode the existence of a third edge intersecting exactly two matching edges. The planarity yields a linear bound on the total degree, which translates into the constant 156 bound.

Negative Result (dimension ≥ 4).

• Theorem 5 shows that the above phenomenon fails in ℝ⁴. For any n there exists a point set P of size Θ(n²) and a family F of n half‑spaces such that every two edges intersect (so ν(H)=1) but any hitting set must contain at least n−½ points (τ(H)=Ω(n)). Thus, affine sign‑rank 4 already permits hypergraphs where ν and τ are arbitrarily far apart.

Algorithmic Consequences.
The “small edge” property yields a simple greedy algorithm: repeatedly pick a small edge, add it to the matching, delete it and all intersecting edges. The resulting matching is guaranteed to be at least 1/156 of the optimum, providing a constant‑factor approximation in polynomial time for maximum matching in the considered geometric hypergraphs (both pseudo‑discs in the plane and half‑spaces in ℝ³).

Connections to ε‑Nets.
The dichotomy directly implies that for pseudo‑discs in ℝ² (including half‑planes) and half‑spaces in ℝ³, ε‑nets of size O(1/ε) exist, matching known bounds. The ℝ⁴ counterexample aligns with recent lower bounds showing that ε‑nets may require Ω((1/ε)·log(1/ε)) points in that setting.

Overall Significance.
The work demonstrates that low affine sign‑rank (≤3) forces a strong combinatorial regularity: matchings and hitting sets are linearly related, and efficient approximation algorithms are possible. Once the dimension reaches four, this regularity collapses, highlighting a sharp threshold for geometric hypergraph complexity. The results have implications for computational geometry, combinatorial optimization, and learning theory, where half‑space classifiers and low‑VC‑dimension concepts are central. Future research may explore intermediate restrictions in four dimensions or seek tighter constants for the matching‑hitting set relationship.