# A randomly weighted minimum spanning tree with a random cost constraint

## TL;DR

This paper analyzes the minimum spanning tree problem on a complete graph with edges having random weights and costs, establishing asymptotic optimal values under a cost constraint for various parameter ranges.

## Contribution

It introduces a novel analysis of the minimum spanning tree problem with random weights and costs under a constraint, deriving asymptotic solutions using duality methods.

## Key findings

- Asymptotic optimal weight values are derived for different parameters.
- The study extends classical MST analysis to incorporate random costs and constraints.
- Dual problem considerations are key to the analysis.

## Abstract

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