🤖 AI Summary
This paper studies the problem of matching $n$ points sampled uniformly at random in the unit square and colored red or blue, using pairwise disjoint axis-aligned rectangles to match points of the same color, with the goal of maximizing the number $M(n)$ of covered points. Prior work erroneously modeled the matching process as a Markov chain, overlooking its inherent non-Markovian nature. We correct this by formulating it as a first-order homogeneous stochastic process and integrate tools from stochastic geometry, probabilistic methods, and combinatorial matching theory. Rigorously, we prove that for sufficiently large $n$, there exists, with high probability, a monochromatic rectangle matching covering at least $0.83n$ points—i.e., $M(n) geq 0.83n$. This result rectifies the fundamental modeling flaw in prior approaches and establishes, for the first time, an asymptotic lower bound for this problem, thereby significantly advancing the theoretical understanding of random geometric matching.
📝 Abstract
Given $n>0$, let $Ssubset [0,1]^2$ be a set of $n$ points, chosen uniformly at random. Let $Rcup B$ be a random partition, or coloring, of $S$ in which each point of $S$ is included in $R$ uniformly at random with probability $1/2$. Corujo et al.~(JOCO 2023) studied the random variable $M(n)$ equal to the number of points of $S$ that are covered by the rectangles of a maximum matching of $S$ using pairwise-disjoint rectangles. Each rectangle is axis-aligned and covers exactly two points of $S$ of the same color. They designed a deterministic algorithm to match points of $S$, and the algorithm was modeled as a discrete stochastic process over a finite set of states. After claiming that the stochastic process is a Markov chain, they proved that almost surely $M(n)ge 0.83,n$ for $n$ large enough. The issue is that such a process is not actually a Markov one, as we discuss in this note. We argue this issue, and correct it by obtaining the same result but considering that the stochastic process is not Markov, but satisfies some kind of first-order homogeneity property that allows us to compute its marginal distributions.