Combinatorial Network Abstraction by Trees and Distances

Abstract

a b s t r a c tWe draw attention to combinatorial network abstraction problems. These are specified by a class P of pattern graphs and a real-valued similarity measure that is based on certain graph properties. For a fixed pattern P and similarity measure , the optimization task on a given graph G is to find a subgraph G ⊆ G which belongs to P and minimizes (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 closenesscentrality 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.

Browsing image database using network spanners

“…However this also points to the challenge in network based visualization -if the network is dense, as similarity based networks often are, the number of edges makes the extraction of truly relevant information difficult [9] and visualization cluttered [10,2,6]. To solve this problem network abstraction or reduction [11] methods construct a sub-network that contains far less data G, E is the set of weighted edges. Wc, and W d are weights based on possibly two different similarity metrics between images.…”