Optimal Random Matchings, Tours, and Spanning Trees in Hierarchically Separated Trees

TL;DR

This paper establishes tight bounds and concentration results for the expected weights of various combinatorial optimization problems, such as matchings, tours, and spanning trees, on random points in hierarchically separated trees.

Contribution

It provides the first tight bounds and concentration results for these problems in the context of hierarchically separated trees, covering multiple problem variants.

Findings

Tight bounds on expected weights for matchings, tours, and spanning trees.

Concentration results for monochromatic problems.

Analysis applicable to both monochromatic and bichromatic cases.

Abstract

We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size $n$ distributed among the leaves of a balanced hierarchically separated tree. We consider {\it monochromatic} and {\it bichromatic} versions of the minimum matching, minimum spanning tree, and traveling salesman problems. We also present tight concentration results for the monochromatic problems.