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Stochastic Minimum Spanning Trees with a Single Sample

Ruben Hoeksma, Gavin Speek, Marc Uetz

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.

Stochastic Minimum Spanning Trees with a Single Sample

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.
Paper Structure (13 sections, 13 theorems, 26 equations, 2 figures)

This paper contains 13 sections, 13 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction, motivation, and model
  2. Notation: Graphs, bonds, minimum spanning trees
  3. Preparatory Results
  4. Single sample minimum spanning tree problem
  5. Exponentially distributed edge weights
  6. Auxiliary results for edge contractions
  7. Upper bound on the performance of sampling based algorithm
  8. Lower bound on the performance of sampling based algorithm
  9. Interpretation as item selection
  10. Extension to matroids
  11. Conclusions
  12. Omitted proofs
  13. Extension to Matroids

Key Result

lemma 1

If $G=(V,E)$ is a connected (multi)graph, then for all edges $e\in{}E$, we have $|B(G/e)|\leq{}|B(G)|$.

Figures (2)

  • Figure 1: Illustration of cut sets and bonds. Red edges indicate the cut sets. Differently colored vertices are different components after removing the cut set.
  • Figure 2: The blue edge is contracted, combining vertices $u$ and $v$ into a single vertex $w$ and creating two parallel edges.

Theorems & Definitions (27)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • lemma 4
  • proof
  • definition 1: SAM
  • theorem 1
  • ...and 17 more