# Probabilistic Analysis of Facility Location on Random Shortest Path Metrics

## TL;DR

This paper analyzes the facility location problem on random shortest path metrics, showing that a simple greedy heuristic is asymptotically optimal for small parameters and providing bounds on its approximation ratio for larger parameters.

## Contribution

It extends probabilistic analysis of facility location algorithms to general metric spaces using random shortest path metrics, beyond Euclidean and non-metric cases.

## Key findings

- Greedy heuristic is asymptotically optimal for small ppa.
- Provides an upper bound on the expected approximation ratio for large ppa.
- Closed-form bounds reduce to ( ext{ln}(n)) or 1 for equal facility costs.

## Abstract

The facility location problem is an NP-hard optimization problem. Therefore, approximation algorithms are often used to solve large instances. Such algorithms often perform much better than worst-case analysis suggests. Therefore, probabilistic analysis is a widely used tool to analyze such algorithms. Most research on probabilistic analysis of NP-hard optimization problems involving metric spaces, such as the facility location problem, has been focused on Euclidean instances, and also instances with independent (random) edge lengths, which are non-metric, have been researched. We would like to extend this knowledge to other, more general, metrics. We investigate the facility location problem using random shortest path metrics. We analyze some probabilistic properties for a simple greedy heuristic which gives a solution to the facility location problem: opening the $\kappa$ cheapest facilities (with $\kappa$ only depending on the facility opening costs). If the facility opening costs are such that $\kappa$ is not too large, then we show that this heuristic is asymptotically optimal. On the other hand, for large values of $\kappa$, the analysis becomes more difficult, and we provide a closed-form expression as upper bound for the expected approximation ratio. In the special case where all facility opening costs are equal this closed-form expression reduces to $O(\sqrt[4]{\ln(n)})$ or $O(1)$ or even $1+o(1)$ if the opening costs are sufficiently small.