D.R. Fulkerson  described a two-phase greedy algorithm to find a minimum cost spanning arborescence and to solve the dual linear program. This was extended by the present author for "kernel systems", a model including the rooted edge-connectivity augmentation problem, as well. A similar type of method was developed by D. Komblum  for "lattice polyhedra", a notion introduced by A. Hoffman and D.E. Schwanz . In order to unify these approaches, here we describe a two-phase greedy algorithm working on a slight extension of lattice polyhedra. This framework includes the rooted node-connectivity augmentation problem, as well, and hence the resulting algorithm, when appropriately specialized, finds a minimum cost of new edges whose addition to a digraph increases its rooted connectivity by one. The only known algorithm for this problem used submodular flows. Actually, the specialized algorithm solves an extension of the rooted edge-connectivity and node-connectivity augmentation problem.
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