Stochastic minimum spanning trees in Euclidean spaces

We study the complexity of geometric minimum spanning trees under a stochastic model of input: Suppose we are given a maste set of points -s ₁, s₂,..., s_n} in d-dimensional Euclidean space, where each point si is active with some independent and arbitrary but known probability p_i. We want to compute the expected length of the minimum spanning tree (MST) of the active points. This particular form of stochastic problems is motivated by the uncertainty inherent in many sources of geometric data but has not been investigated before in computational geometry to the best of our knowledge. Our main results include the following. (1). We show that the stochastic MST problem is #P-hard for any dimension d ≥ 2.( 2). We present a simple fully polynomial randomized approximation scheme (FPRAS) for a metric space, and thus also for any Euclidean space. (3). For d = 2, we present two deterministic approximation algorithms: an O(n⁴)-time constant-factor algorithm, and a PTAS based on a combination of shifted quadtrees and dynamic programming. (4). We show that in a general metric space the tail bounds of the distribution of the MST length cannot be approximated to any multiplicative factor in polynomial time under the assumption that P ≠ NP. In addition to this existential model of stochastic input, we also briefly consider a locational model where each point is present with certainty but its location is probabilistic.