A Simple Proof of the $O( \sqrt{n} \log^{3 / 4} n )$ Upright Matching Bound

Abstract

The stochastic upright matching problem has had many important applications, most notably in statistics and the average-case analysis of algorithms. A problem instance is a set of n points chosen uniformly at random in the unit square. The points are labeled with signs; the signs are chosen independently and each is equally likely to be a plus or minus. An up-right matching of S is a matching of minus points to plus points such that if $( x,y )$ is a minus point matched to the plus point $( x',y' )$, then $x\leqq x'$ and $y\leqq y'$. The problem is to estimate the expected number of points left unmatched in a maximum upright matching of S. It is well known that if $U_n $ denotes the number of unmatched points, then $E[ U_n ] = \Theta ( \sqrt{n} \log^{3/4} n )$. Existing proofs of the upper bound $O( \sqrt{n} \log^{3/4} n )$ are quite long and difficult to follow. This paper presents a much simpler and more compact proof. A distinctive feature of the new proof is the use of Fourier expansions.