Primal-dual approach for directed vertex connectivity augmentation and generalizations

In their seminal paper, Frank and Jord´an [1995] show that a large class of optimization problems, including certain directed graph augmentation, fall into the class of covering supermodular functions over pairs of sets. They also give an algorithm for such problems, however, it relies on the ellipsoid method. Prior to our result, combinatorial algorithms existed only for the 0–1 valued problem. Our key result is a combinatorial algorithm for the general problem that includes directed vertex or S−T connectivity augmentation. The algorithm is based on Bencz´ur’s previous algorithm for the 0–1 valued case [Bencz´ur 2003]. Our algorithm uses a primal-dual scheme for finding covers of partially ordered sets that satisfy natural abstract properties as in Frank and Jord´an. For an initial (possibly greedy) cover, the algorithm searches for witnesses for the necessity of each element in the cover. If no two (weighted) witnesses have a common cover, the solution is optimal. As long as this is not the case, the witnesses are gradually exchanged for smaller ones. Each witness change defines an appropriate change in the solution; these changes are finally unwound in a shortest-path manner to obtain a solution of size one less.