1. An approximation method carried out by computer readable program code operable by a computer system for designing a minimum cost network from defined available network elements having values for cost, required pairs, connectivity requirements and penalty costs for insufficient connectivity by selecting from these available elements, comprising: constructing a linear programming (LP) relaxation for a graph having edges and vertices with values corresponding to those of the available network elements; finding an optimal basic solution to the LP; applying a selection criterion to penalty costs and edge costs identified in the optimal basic solution and including the selected penalty costs and edges in the minimum cost network; constructing a residual LP relaxation including remaining pairs with unsatisfied connectivity and edges; and fixing selected edge coefficient values to 1 in the residual LP; repeating the steps of the method to include additional penalty costs and edge costs in the minimum cost network until all connectivity requirements are satisfied.