Stochastic Minimum Spanning Trees with a Single Sample

TL;DR

This paper investigates the stochastic minimum spanning tree problem with only a single sample per edge, providing performance guarantees and tight bounds for exponential distributions and general matroids.

Contribution

It introduces a performance analysis of a sampling-based algorithm for stochastic MSTs with single samples, extending results to arbitrary matroids.

Findings

Performance guarantee equals the largest bond in the graph for exponential weights.

Guarantee is tight for all graphs.

Results extend to arbitrary matroids with performance tied to largest co-circuit.

Abstract

We consider the minimum spanning tree problem in a setting where the edge weights are stochastic from unknown distributions, and the only available information is a single sample of each edge's weight distribution. In this setting, we analyze the expected performance of the algorithm that outputs a minimum spanning tree for the sampled weights. We compare to the optimal solution when the distributions are known. For every graph with weights that are exponentially distributed, we show that the sampling based algorithm has a performance guarantee that is equal to the size of the largest bond in the graph. Furthermore, we show that for every graph this performance guarantee is tight. The proof is based on two separate inductive arguments via edge contractions, which can be interpreted as reducing the spanning tree problem to a stochastic item selection problem. We also generalize these results to arbitrary matroids, where the performance guarantee is equal to the size of the largest co-circuit of the matroid.