Randomization Helps Computing a Minimum Spanning Tree under Uncertainty

We consider the problem of finding a minimum spanning tree (MST) in a graph with uncertain edge weights given by open intervals on the edges. The exact weight of an edge in the corresponding uncertainty interval can be queried at a given cost. The task is to determine a possibly adaptive query sequence of minimum total cost for finding an MST. For uniform query cost, a deterministic algorithm with best possible competitive ratio 2 is known [7].

We solve a long-standing open problem by showing that randomized query strategies can beat the best possible competitive ratio 2 of deterministic algorithms. Our randomized algorithm achieves expected competitive ratio

$1+1/\sqrt{2}\approx 1.707$

. This result is based on novel structural insights to the problem enabling an interpretation as a generalized online bipartite vertex cover problem. We also consider arbitrary, edge-individual query costs and show how to obtain algorithms matching the best known competitive ratios for uniform query cost. Moreover, we give an optimal algorithm for the related problem of computing the exact weight of an MST at minimum query cost. This algorithm is based on an interesting relation between different algorithmic approaches using the cycle-property and the cut-property characterizing MSTs. Finally, we argue that all our results also hold for the more general setting of matroids. All our algorithms run in polynomial time.