Random Shortest Paths: Non-Euclidean Instances for Metric Optimization Problems

TL;DR

This paper investigates the structural properties of random shortest path metrics in non-Euclidean complete graphs and analyzes the performance of heuristics for classic optimization problems, revealing these instances are generally easier than worst-case scenarios.

Contribution

It introduces a new structural analysis of non-Euclidean random shortest path metrics and applies it to evaluate heuristic approximation ratios and running times.

Findings

Heuristic algorithms perform better on these random instances than worst-case bounds suggest.

Constructed good clusterings in the metric space facilitate analysis.

Random shortest path metrics are typically easier for optimization heuristics.

Abstract

Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtained as follows: Every edge of a complete graph gets a weight drawn independently at random. The distance between two nodes is then the length of a shortest path (with respect to the weights drawn) that connects these nodes. We prove structural properties of the random shortest path metrics generated in this way. Our main structural contribution is the construction of a good clustering. Then we apply these findings to analyze the approximation ratios of heuristics for matching, the traveling salesman problem (TSP), and the k-median problem, as well as the running-time of the 2-opt heuristic for the TSP. The bounds that we obtain are considerably better than the respective worst-case bounds. This suggests that random shortest path metrics are easy instances, similar to random Euclidean instances, albeit for completely different structural reasons.