It is difficult to tell if there is a Condorcet spanning tree

We apply the well-known Condorcet criterion from voting theory outside of its classical framework and link it with spanning trees of an undirected graph. In situations in which a network, represented by a spanning tree of an undirected graph, needs to be installed, decision-makers typically do not agree on the network to be implemented. Instead, each of these decision-makers has her own ideal conception of the network. In order to derive a group decision, i.e., a single spanning tree for the entire group of decision-makers, the goal would be a spanning tree that beats each other spanning tree in a simple majority comparison. When comparing two dedicated spanning trees, a decision-maker will be considered to be more satisfied with the one that is “closer” to her proposal. In this context, the most basic and natural measure of distance is the usual set difference: we simply count the number of edges the spanning tree has in common with the proposal of the decision-maker. In this work, we show that it is computationally intractable to decide (1) if such a spanning tree exists, and (2) if a given spanning tree satisfies the Condorcet criterion.