Edge-connectivity augmentation preserving simplicity

Given a simple graph G = (V, E), our goal is to find a smallest set F of new edges such that G = (V, E boolean OR F) is k-edge-connected and simple. Recently this problem was shown to be NP-complete. In this paper we prove that if OPTPk is high enough-depending on k only-then OPTSk = OPTPk holds, where OPTSk (OPTPk) is the size of an optimal solution of the augmentation problem with (without) the simplicity-preserving requirement, respectively. Furthermore, OPTSk - OPTPk less than or equal to g(k) holds for a certain (quadratic) function of k. Based on these facts an algorithm is given which computes an optimal solution in time O(n(4)) for any fixed k. Some of these results are extended to the case of nonuniform demands as well.