Online Minimum Spanning Trees with Weight Predictions

TL;DR

This paper studies online algorithms for minimum spanning trees with weight predictions, introducing error-based performance analysis and demonstrating the optimality of a simple approach while proposing a greedy variant with improved insights.

Contribution

It introduces a new error measure for algorithms with predictions, proves the optimality of Follow-the-Predictions, and presents a novel greedy algorithm with a first random order analysis.

Findings

Follow-the-Predictions is optimal under competitive analysis.

The greedy variant outperforms the simple algorithm in certain scenarios.

First random order analysis of a non-trivial online algorithm with predictions.

Abstract

We consider the minimum spanning tree problem with predictions, using the weight-arrival model, i.e., the graph is given, together with predictions for the weights of all edges. Then the actual weights arrive one at a time and an irrevocable decision must be made regarding whether or not the edge should be included into the spanning tree. In order to assess the quality of our algorithms, we define an appropriate error measure and analyze the performance of the algorithms as a function of the error. We prove that, according to competitive analysis, the simplest algorithm, Follow-the-Predictions, is optimal. However, intuitively, one should be able to do better, and we present a greedy variant of Follow-the-Predictions. In analyzing that algorithm, we believe we present the first random order analysis of a non-trivial online algorithm with predictions, by which we obtain an algorithmic separation. This may be useful for distinguishing between algorithms for other problems when Follow-the-Predictions is optimal according to competitive analysis.