On Triangular Separation of Bichromatic Point Sets

TL;DR

The paper studies the minimum number of triangles needed to separate blue and red points in the plane, introducing Overlap-Separation and Disjoint-Separation variants and reducing the problem to blue-coverage (and similarly for red) subproblems. It provides a rigorous NP-hardness result by reducing Planar Monotone $3$-SAT to the blue-overlap and blue-no-overlap cases via a gadget-based construction, showing that a satisfiable instance corresponds to a feasible triangle cover of size $k+2m$. Beyond hardness, the authors derive worst-case bounds on the required number of triangles across multiple settings, using geometric constructions and algorithmic techniques such as sweeping and convex-hull analysis, and they complement these results with computer-assisted small-$n$ experiments (e.g., SAT formulations and exact counts). The work thereby advances understanding of the computational limits for geometric separation with simple shapes and provides conjectures and data motivating future tight bounds and algorithmic approaches.

Abstract

We address the problem of computing the minimum number of triangles to separate a set of blue points from a set of red points in $\mathbb{R}^2$. A set of triangles is a \emph{separator} of one color from the other if every point of that color is contained in some triangle and no triangle contains points of both colors. We consider several variants of the problem depending on whether the triangles are allowed to overlap or not and whether all points or just the blue points need to be contained in a triangle. We show that computing the minimum cardinality triangular separator of a set of blue points from a set of red points is NP-hard and further investigate worst case bounds on the minimum cardinality of triangular separators for a bichromatic set of $n$ points.

Figures (7)

• Figure 1: The covering triangles of a variable gadget $x_i$. Robert: The white space between the dashed lines can be used for red points.
• Figure 2: Planar embedding of the PM-3SAT formula $\phi$.
• Figure 3: The covering triangles of a variable gadget $x_i$.
• Figure 4: An overview of the covering triangles for clause $c_j$.
• Figure 5: A visualization for the proof of \ref{['prop:induct+comp']}.

Theorems & Definitions (33)

• Conjecture 1
• Conjecture 2
• Conjecture 3
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• theorem 1