Combinatorial network abstraction by trees and distances

Abstract

We draw attention to combinatorial network abstraction problems. These are specified by a class P of pattern graphs and a real-valued similarity measure Q that is based on certain graph properties. For a fixed pattern P and similarity measure Q, the optimization task on a given graph G is to find a subgraph G' subset of G which belongs to P and minimizes Q(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-infinity. 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.