k-edge-connectivity augmentation problem with upper bounds on edge multiplicity

The k-edge-connectivity augmentation problem of a graph with upper bounds on edge multiplicity is defined as follows: “Given a graph G=(V, E) and a function f: V×V→Z⁺ (nonnegative integers), find an edge set of minimum cardinality among those sets E´ such that G´=(V, E∪E´) is k-edge-connected and, for any pair of vertices u, v, the number of edges added between u and v is no more than f(u, v).” In this paper, we first show that this problem is NP-complete. Then we propose three heuristic algorithms and compare their capability through experiment