The Planted Spanning Tree Problem

2025-02-12

AI Summary

This paper studies detection and recovery of a hidden planted spanning tree—either a uniform random spanning tree or a Hamiltonian path—in a weighted complete graph: planted edge weights are i.i.d. from a fixed distribution $P$, while non-planted edge weights follow a distribution $Q_n$ satisfying $lim_{n oinfty} n Q'_n(0) = 1$. Using the minimum spanning tree (MST) algorithm, we derive for the first time an asymptotic fixed-point equation characterizing the edge recovery rate. We extend Frieze’s classical result—where the expected MST weight converges to $zeta(3)$ in the homogeneous Erdős–Rényi model—to this inhomogeneous weighted setting with hidden structure, and precisely determine the limiting average MST weight under planting. Furthermore, we construct a statistically optimal hypothesis test based on the total MST weight, achieving strong consistency: both Type-I and Type-II error probabilities vanish as $n o infty$.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the problem of detecting and recovering a planted spanning tree $M_n^*$ hidden within a complete, randomly weighted graph $G_n$. Specifically, each edge $e$ has a non-negative weight drawn independently from $P_n$ if $e in M_n^*$ and from $Q_n$ otherwise, where $P_n equiv P$ is fixed and $Q_n$ scales with $n$ such that its density at the origin satisfies $lim_{n oinfty} n Q'_n(0)=1.$ We consider two representative cases: when $M_n^*$ is either a uniform spanning tree or a uniform Hamiltonian path. We analyze the recovery performance of the minimum spanning tree (MST) algorithm and derive a fixed-point equation that characterizes the asymptotic fraction of edges in $M_n^*$ successfully recovered by the MST as $n o infty.$ Furthermore, we establish the asymptotic mean weight of the MST, extending Frieze's $zeta(3)$ result to the planted model. Leveraging this result, we design an efficient test based on the MST weight and show that it can distinguish the planted model from the unplanted model with vanishing testing error as $n o infty.$ Our analysis relies on an asymptotic characterization of the local structure of the planted model, employing the framework of local weak convergence.

Problem

• Detect and recover planted spanning trees
• Analyze MST algorithm's recovery performance
• Design efficient test for model distinction

Innovation

• Minimum Spanning Tree algorithm
• Fixed-point equation analysis
• Local weak convergence framework