Short Note on Edge Connectivity Augmentation 22 Augmenting Acyclic

Consider a network N = (V; Ec; c) with an integer valued capacity function c: V V ! Z+, and let k be a positive integer. What is the minimal total increase by which the individual capacities must be increased such that the edge connectivity number is at least k? Clearly the defect r(u) := max(k ? c(u); 0) summing over all vertices, called T is a lower bound for. Kajitani and Ueno 7] proved that if (V; Ec) is a tree then = T. We extend this result to the larger class of acyclic digraphs. Frank 4] gives a min-max formula for which is proved using Maders 8] splitting theorem. In order to obtain an eecient implementation of the resulting strong polynomial time algorithm, one must carry out some reduction and splitting operations which, in turn, entail performing several maximal ow computations. We give an implementation which signiicantly decreases the time complexity of the reduction phase, and substantially reduces the running time of the entire algorithm. Furthermore we give some computational results.