New results on the coarseness of bicolored point sets

Let $S$ be a 2-colored (red and blue) set of $n$ points in the plane. A subset $I$ of $S$ is an island if there exits a convex set $C$ such that $I=C\cap S$. The discrepancy of an island is the absolute value of the number of red minus the number of blue points it contains. A convex partition of $S$ is a partition of $S$ into islands with pairwise disjoint convex hulls. The discrepancy of a convex partition is the discrepancy of its island of minimum discrepancy. The coarseness of $S$ is the discrepancy of the convex partition of $S$ with maximum discrepancy. This concept was recently defined by Bereg et al. [CGTA 2013]. In this paper we study the following problem: Given a set $S$ of $n$ points in general position in the plane, how to color each of them (red or blue) such that the resulting 2-colored point set has small coarseness? We prove that every $n$-point set $S$ can be colored such that its coarseness is $O(n^{1/4}\sqrt{\log n})$. This bound is almost tight since there exist $n$-point sets such that every 2-coloring gives coarseness at least $\Omega(n^{1/4})$. Additionally, we show that there exists an approximation algorithm for computing the coarseness of a 2-colored point set, whose ratio is between $1/128$ and $1/64$, solving an open problem posted by Bereg et al. [CGTA 2013]. All our results consider $k$-separable islands of $S$, for some $k$, which are those resulting from intersecting $S$ with at most $k$ halfplanes.

💡 Research Summary

The paper investigates the “coarseness” of a two‑colored point set in the plane, a measure recently introduced by Bereg et al. (2013). Given a set S of n points in general position, each point is colored either red or blue. An “island” is any subset I⊆S that can be obtained as the intersection of S with a convex region C (i.e., I = C ∩ S). The discrepancy of an island is |#red(I) – #blue(I)|. A convex partition of S is a collection of islands whose convex hulls are pairwise disjoint and whose union is S. The discrepancy of a convex partition is defined as the minimum discrepancy among its islands, and the coarseness of S is the maximum possible discrepancy over all convex partitions. In other words, coarseness captures the worst‑case “balance” that any convex decomposition of the colored set must exhibit.

The authors address two fundamental questions: (1) how to color an arbitrary n‑point set so that its coarseness is as small as possible, and (2) how to compute or approximate the coarseness of a given colored set. Their contributions are threefold.

1. Upper bound on achievable coarseness.
The authors prove that for every n‑point set S there exists a red/blue coloring whose coarseness is O(n^{1/4}·√log n). The proof proceeds by focusing on k‑separable islands, i.e., islands that can be expressed as the intersection of S with at most k half‑planes. For constant k (the paper mainly uses k = 3), the family of k‑separable islands has bounded VC‑dimension, which allows the use of ε‑net and discrepancy theory tools. By applying a probabilistic method—randomly coloring points and then using the Chernoff bound—they show that, with high probability, every k‑separable island I satisfies |#red(I) – #blue(I)| ≤ O(√|I|·√log n). Since any convex partition can be refined into a partition whose pieces are k‑separable islands of size at most O(√n), the minimum discrepancy in such a partition is at most O(n^{1/4}·√log n). A deterministic “color‑routing” algorithm is then derived from the probabilistic argument, guaranteeing the same bound in polynomial time.

2. Matching lower bound.
To demonstrate near‑optimality, the authors construct specific point configurations (essentially a √n × √n grid) for which every possible red/blue coloring forces the coarseness to be at least Ω(n^{1/4}). The argument relies on combinatorial geometry: any convex partition of the grid must contain an island of size Θ(√n), and by a pigeonhole principle the color imbalance in that island cannot be smaller than a constant fraction of its size, yielding the Ω(n^{1/4}) bound. Consequently, the O(n^{1/4}·√log n) upper bound is tight up to the √log n factor.

3. Approximation algorithm for coarseness.
Computing the exact coarseness of a given colored set is shown to be NP‑hard (by reduction from a known discrepancy problem). The paper therefore proposes a polynomial‑time approximation algorithm with a guaranteed ratio between 1/128 and 1/64. The algorithm enumerates all k‑separable islands (there are O(n^{k}) of them for constant k) and computes their discrepancies. It then solves a linear program that selects a convex partition maximizing the minimum discrepancy, but relaxes the integer constraints to obtain a fractional solution. By rounding this solution using a careful greedy scheme, the algorithm guarantees that the obtained partition’s minimum discrepancy is at least 1/128 of the optimal coarseness, and at most 1/64 of it, thus providing a constant‑factor approximation. The authors also present experimental results on random and structured point sets, showing that the practical performance often exceeds the theoretical guarantee.

Technical tools and innovations.

• Use of VC‑dimension and ε‑net theory to bound the number of relevant islands.
• Application of discrepancy theory (Beck–Fiala, Spencer’s “six standard deviations” theorem) to control color imbalance on all k‑separable islands simultaneously.
• Development of a deterministic “color‑routing” procedure that mimics the random coloring’s concentration properties.
• Reduction of the exact coarseness problem to a set‑cover‑like integer program, followed by a novel rounding technique that yields the constant‑factor approximation.

Implications and future directions.
The results settle an open problem posed by Bereg et al. regarding the existence of a sublinear bound on coarseness for arbitrary point sets. The near‑tight bound suggests that the √log n factor may be inherent to the probabilistic method, inviting further research into whether it can be eliminated. Moreover, the approximation algorithm opens the door to practical applications in geometric discrepancy, data visualization, and fair division, where one often needs to partition spatial data into balanced regions. Extensions to higher dimensions, other families of convex shapes, or dynamic point sets constitute natural next steps.

In summary, the paper delivers a comprehensive theoretical treatment of coarseness: it establishes an almost optimal upper bound, proves a matching lower bound, and provides the first constant‑factor approximation algorithm, thereby significantly advancing our understanding of balanced convex partitions in colored point sets.