On the random 2‐stage minimum spanning tree

It is known [A. M. Frieze, Discrete Appl Math 10 (1985), 47–56] that if the edge costs of the complete graph K n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to $\zeta(3)=\sum_{i=1}^{\infty}i^{-3}$ . Here we consider the following stochastic two‐stage version of this optimization problem. There are two sets of edge costs c M : E → ℝ and c T : E → ℝ, called Monday's prices and Tuesday's prices, respectively. For each edge e , both costs c M ( e ) and c T ( e ) are independent random variables, uniformly distributed in [0, 1]. The Monday costs are revealed first. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price c M ( e ), or to wait until its Tuesday price c T ( e ) appears. The set of edges X M bought on Monday is then completed by the set of edges X T bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost ζ(3)/2 + o (1). We show that, in the case of two‐stage optimization, the expected value of the optimal cost exceeds ζ(3)/2 by an absolute constant ε > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than α and completes them on Tuesday in an optimal way, and show that the optimal choice for α is α = 1/n with the expected cost ζ(3) − 1/2 + o(1). The threshold heuristic is shown to be sub‐optimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning out‐arborescence rooted at a fixed vertex r , and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 − 1/ e + o (1). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006