# A randomly weighted minimum arborescence with a random cost constraint

## TL;DR

This paper analyzes the asymptotic behavior of the minimum arborescence problem with random weights and costs under a constraint, using duality to derive results for various parameter ranges.

## Contribution

It introduces a probabilistic model for the minimum arborescence with random weights and costs, and characterizes the asymptotic optimal weight considering a cost constraint.

## Key findings

- Asymptotic optimal weight derived for different parameter ranges
- Dual problem approach used to analyze the constrained arborescence
- Results applicable to complete digraphs with uniform random edge attributes

## Abstract

We study the minimum spanning arborescence problem on the complete digraph $\vec{K}_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent uniform random variable $U^\alpha$ where $\alpha\leq 1$ and $U$ is uniform $[0,1]$. There is also a constraint that the spanning arborescence $T$ must satisfy $C(T)\leq c_0$. We establish, for a range of values for $c_0,\alpha$, the asymptotic value of the optimum weight via the consideration of a dual problem.