Advice complexity of online non-crossing matching

We study online matching in the Euclidean 2-dimensional plane with the non-crossing constraint. The offline version was introduced by Atallah in 1985 and the online version was introduced and studied more recently by Bose et al. The input to the monochromatic non-crossing matching (MNM) problem consists of a sequence of points. Upon arrival of a point, an algorithm can decide to match it with a previously unmatched point or leave it unmatched. The line segments corresponding to the edges in the matching should not cross each other, and the goal is to maximize the size of the matching. The decisions are irrevocable, and while an optimal offline solution always matches all the points, an online algorithm cannot match all the points in the worst case, unless it is given some additional information, i.e., advice. In the bichromatic version (BNM), blue points are given in advance and the same number of red points arrive online. The goal is to maximize the number of red points matched to blue points without creating any crossings.

We show that the advice complexity of solving BNM optimally on a circle (or, more generally, on inputs in a convex position) is tightly bounded by the logarithm of the $n^{\text{th}}$ Catalan number from above and below. This result corrects the previous claim of Bose et al. that the advice complexity is $\log (n!)$. At the heart of the result is a connection between the non-crossing constraint in online inputs and the 231-avoiding property of permutations of n elements. We also show a lower bound of $n/3-1$ and an upper bound of 3n on the advice complexity for MNM on a plane. This gives an exponential improvement over the previously best-known lower bound and an improvement in the constant of the leading term in the upper bound. In addition, we establish a lower bound of $\frac{\alpha }{2}D\left(\frac{2(1-\alpha )}{\alpha }\right||1/4)n$ on the advice complexity for achieving competitive ratio $\alpha \in (16/17,1)$ for MNM on a circle where $D\left(p\right|\left|q\right)$ is the relative entropy between two Bernoulli random variables with parameters p and q. Standard tools from advice complexity, such as partition trees and reductions from string guessing problems, do not seem to apply to MNM/BNM, so we have to design our lower bounds from first principles.