  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.
