The Planted Spanning Tree Problem

TL;DR

This paper investigates the detection and recovery of a hidden spanning tree within a complete graph with random weights, analyzing the performance of MST-based algorithms and establishing thresholds for successful identification.

Contribution

It introduces a fixed-point characterization of the recovery fraction, extends Frieze's MST weight result to the planted setting, and proposes an efficient test to distinguish planted from unplanted models.

Findings

MST algorithm's recovery performance is characterized by a fixed-point equation.

The asymptotic mean weight of the MST is extended to the planted model.

An efficient MST-based test can reliably distinguish the planted model as n grows large.

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 $\lim_{n\to\infty} n Q'_n(0)=1.$ 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 $n \to \infty.$ 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 $n \to \infty.$ Our analysis relies on an asymptotic characterization of the local structure of the planted model, employing the framework of local weak convergence.