Variable Neighborhood Search for the Generalized Minimum Edge Biconnected Network Problem

The Generalized Minimum Edge Biconnected Network Problem (GMEBCNP) is defined as follows. We consider an undirected weighted graph G = 〈V, E, c〉 with node set V , edge set E, and edge cost function c : E → R. Node set V is partitioned into r pairwise disjoint clusters V1, V2, . . . , Vr, ⋃r i=1 Vi = V, Vi ∩ Vj = ∅ ∀i, j = 1, . . . , r, i 6= j. A solution to the GMEBCNP defined on G is a subgraph S = 〈P, T 〉, P = {p1, . . . , pr} ⊆ V connecting exactly one node from each cluster, i.e. pi ∈ Vi, ∀i = 1, . . . , r, and containing no bridges [2, 7, 8], see Figure 1. A bridge is an edge which does not lie on any cycles and thus its removal would disconnect the graph. The costs of such an edge biconnected network are its total edge costs, i.e. c(T ) = ∑ (u,v)∈T c(u, v), and the objective is to identify a solution with minimum costs. This problem obviously is NP hard since already the task of finding a minimum cost biconnected network spanning all nodes of a given graph is NP hard.