Approximate Tree Completion and Learning-Augmented Algorithms for Metric Minimum Spanning Trees

TL;DR

This paper presents a practical framework for approximating metric MSTs efficiently by combining heuristics and learning-augmented algorithms, achieving near-optimal solutions faster than traditional methods.

Contribution

It introduces a novel two-step approach for metric MSTs, including a subquadratic approximation algorithm and a learning-augmented method that leverages overlap with the optimal MST.

Findings

Achieves a 2.62-approximation in subquadratic time.

Practically finds nearly optimal spanning trees across various metrics.

Significantly faster than exact algorithms while maintaining high quality.

Abstract

Finding a minimum spanning tree (MST) for $n$ points in an arbitrary metric space is a fundamental primitive for hierarchical clustering and many other ML tasks, but this takes $\Omega(n^2)$ time to even approximate. We introduce a framework for metric MSTs that first (1) finds a forest of disconnected components using practical heuristics, and then (2) finds a small weight set of edges to connect disjoint components of the forest into a spanning tree. We prove that optimally solving the second step still takes $\Omega(n^2)$ time, but we provide a subquadratic 2.62-approximation algorithm. In the spirit of learning-augmented algorithms, we then show that if the forest found in step (1) overlaps with an optimal MST, we can approximate the original MST problem in subquadratic time, where the approximation factor depends on a measure of overlap. In practice, we find nearly optimal spanning trees for a wide range of metrics, while being orders of magnitude faster than exact algorithms.