On the random 2-stage minimum spanning tree

It is known 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 ζ(3) = Σ _i=1 ^∞ 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 a and completes them on Tuesday in an optimal way, and show that the optimal choice for a 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).