Mathematical programming models for some smallest-world problems

Given a weighted graph G, in the minimum-cost-edge-selection problem (MCES), a minimum weighted set of edges is chosen subject to an upper bound on the diameter of graph G. Similarly, in the minimum-diameter-edge-selection problem (MDES), a set of edges is chosen to minimize the diameter subject to an upper bound on their total weight. These problems are shown to be equivalent and proven to be NP-complete. MCES is then formulated as a 0-1 integer programming problem. The problems MCES and MDES provide models for determining smallest-world networks and for measuring the “small-worldness” of graphs.