Combinatorial network abstraction by trees and distances

We draw attention to combinatorial network abstraction problems. These are specified by a class P of pattern graphs and a real-valued similarity measure @r that is based on certain graph properties. For a fixed pattern P and similarity measure @r, the optimization task on a given graph G is to find a subgraph G^'@?G which belongs to P and minimizes @r(G,G^'). In this work, we consider this problem for the natural and somewhat general case of trees and distance-based similarity measures. In particular, we systematically study spanning trees of graphs that minimize distances, approximate distances, and approximate closeness-centrality with respect to standard vector- and matrix-norms. Complexity analysis within a unifying framework shows that all considered variants of the problem are NP-complete, except for the case of distance-minimization with respect to the norm L"~. If a subset of edges can be ''forced'' into the spanning tree, no polynomial-time constant-factor approximation algorithmexists for the distance-approximation problems unless P=NP.