On the random 2-stage minimum spanning tree

TLDR: The directed version of the problem is discussed, where the task is to construct a spanning out‐arborescence rooted at a fixed vertex r, and it is shown that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 − 1/e + o(1).

Abstract: It is known [7] that if the edge costs of the complete graph Kn 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=1i-3. Here we consider the following stochastic two-stage version of this optimization problem. There are two sets of edge costs cM: E ← R and cT: E ← R, called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs cM(e) and cT(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 cM(e), or to wait until its Tuesday price cT(e) appears. The set of edges XM bought on Monday is then completed by the set of edges XT 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 ∈ > 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).