# The Dyck bound in the concave 1-dimensional random assignment model

## TL;DR

This paper introduces the Dyck matching configuration for one-dimensional random assignment with concave cost functions, computes its average cost asymptotics, and conjectures its optimality, supported by numerical evidence.

## Contribution

It proposes the Dyck matching as a new configuration for concave cost assignment problems and derives its asymptotic average cost, suggesting it matches the optimal cost.

## Key findings

- Average cost scales as N, N log N, or N^{1/2 + p} depending on p.
- Dyck matching's average cost conjectured to match the optimal cost.
- Numerical data supports the conjecture.

## Abstract

We consider models of assignment for random $N$ blue points and $N$ red points on an interval of length $2N$, in which the cost for connecting a blue point in $x$ to a red point in $y$ is the concave function $|x-y|^p$, for $0<p<1$. Contrarily to the convex case $p>1$, where the optimal matching is trivially determined, here the optimization is non-trivial. The purpose of this paper is to introduce a special configuration, that we call the \emph{Dyck matching}, and to study its statistical properties. We compute exactly the average cost, in the asymptotic limit of large $N$, together with the first subleading correction. The scaling is remarkable: it is of order $N$ for $p<\frac{1}{2}$, order $N \ln N$ for $p=\frac{1}{2}$, and $N^{\frac{1}{2}+p}$ for $p>\frac{1}{2}$, and it is universal for a wide class of models. We conjecture that the average cost of the Dyck matching has the same scaling in $N$ as the cost of the optimal matching, and we produce numerical data in support of this conjecture. We hope to produce a proof of this claim in future work.