Finding minimum spanning trees via local improvements

TL;DR

This paper analyzes a local search algorithm for finding minimum spanning trees on complete graphs with random edge weights, identifying a threshold parameter that determines the algorithm's success probability.

Contribution

It introduces a parameterized local search method for MSTs and characterizes a threshold value dictating its effectiveness on random weighted graphs.

Findings

Identifies a threshold $ ho^*$ for local search success

Shows high probability of success when $ ho > ho^*$

Demonstrates failure with high probability when $ ho < ho^*$

Abstract

We consider a family of local search algorithms for the minimum-weight spanning tree, indexed by a parameter $\rho$. One step of the local search corresponds to replacing a connected induced subgraph of the current candidate graph whose total weight is at most $\rho$ by the minimum spanning tree (MST) on the same vertex set. Fix a non-negative random variable $X$, and consider this local search problem on the complete graph $K_n$ with independent $X$-distributed edge weights. Under rather weak conditions on the distribution of $X$, we determine a threshold value $\rho^*$ such that the following holds. If the starting graph (the "initial candidate MST") is independent of the edge weights, then if $\rho > \rho^*$ local search can construct the MST with high probability (tending to $1$ as $n \to \infty$), whereas if $\rho < \rho^*$ it cannot with high probability.