Random assignment problems on ${2d}$ manifolds

TL;DR

This paper analyzes the assignment problem on 2D manifolds, deriving exact corrections to the average cost using spectral properties of the Laplace--Beltrami operator, supported by numerical validation.

Contribution

It introduces a method to compute the first manifold-dependent correction to the assignment problem's cost using spectral analysis within a field-theoretical framework.

Findings

Exact computation of the constant correction term for various surfaces.

Validation of theoretical predictions through extensive numerical experiments.

Demonstration of the spectral approach's effectiveness in geometric optimization problems.

Abstract

We consider the assignment problem between two sets of $N$ random points on a smooth, two-dimensional manifold $\Omega$ of unit area. It is known that the average cost scales as $E_{\Omega}(N)\sim\frac{1}{2\pi}\ln N$ with a correction that is at most of order $\sqrt{\ln N\ln\ln N}$. In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $\Omega$-dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace--Beltrami operator on $\Omega$. We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.