Abstract
We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set of points in the plane $${\mathcal {S}} \subset {\mathbb {R}}^2$$ can be 2-colored such that every axis-parallel square that contains at least m points from $${\mathcal {S}}$$ contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering 2-coloring points with respect to homothets of a fixed parallelogram.
Highlights
In this paper we consider the problem of coloring a given set of points in the plane such that every region from a given set of regions contains a point from each color class
For a long time no positive results were known about cover-decomposability and geometric hypergraph coloring problems concerning homothets of a given shape
We have presented a general framework showing that if a convex polygon P satisfies a certain property, there is an absolute constant m that depends only on P such that every set of points in the plane can be 2-colored such that every homothet of P that contains at least m points contains points of both colors
Summary
In this paper we consider the problem of coloring a given set of points in the plane such that every region from a given set of regions contains a point from each color class. For a long time no positive results were known about cover-decomposability and geometric hypergraph coloring problems concerning homothets of a given shape. An important tool for obtaining these results is the notion of self-coverability (see Section 2.2), which is essential for proving our results It is an interesting open problem whether mk = O(k) and m∗k = O(k) for the homothets of a given triangle. Given an (open or closed) parallelogram Q and a finite set S of points in the plane, the points of S can be 2-colored in polynomial time, such that any homothet of Q that contains at least mq points contains points of both colors This is the first example that exhibits such different behavior for coloring the primal and dual hypergraphs with respect to the family of some geometric regions. Some proofs are omitted and can be found in the full version of this extended abstract
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