Stochastic Minimum Spanning Trees and Related Problems

We investigate the computational complexity of minimum spanning trees and maximum flows in a simple model of stochastic networks, where each node or edge of an undirected master graph can fail with an independent and arbitrary probability. We show that computing the expected length of the MST or the value of the max-flow is #P-Hard, but that for the MST it can be approximated within O(log n) factor for metric graphs. The hardness proof for the MST applies even to Euclidean graphs in 3 dimensions. We also show that the tail bounds for the MST cannot be approximated in general to any multiplicative factor unless P = NP. This stochastic MST problem was mentioned but left unanswered by Bertsimas, Jaillet and Odoni [Operations Research, 1990] in their work on a priori optimization. More generally, we also consider the complexity of linear programming under probabilistic constraints, and show it to be #P-Hard. If the linear program has a constant number of variables, then it can be solved exactly in polynomial time. For general dimensions, we give a randomized algorithm for approximating the probability of LP feasibility.