Finding minimum spanning trees via local improvements

Abstract

We consider a family of local search algorithms for the minimum-weight spanning tree, indexed by a parameter ρ. One step of the local search corresponds to replacing a connected induced subgraph of the current candidate graph whose total weight is at most ρ 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 Kn with independent X-distributed edge weights. Under rather weak conditions on the distribution of X, we determine a threshold value ρ * such that the following holds. If the starting graph (the "initial candidate MST") is independent of the edge weights, then if ρ > ρ * local search can construct the MST with high probability (tending to 1 as n → ∞), whereas if ρ < ρ * it cannot with high probability.