Greedoids from flames

A digraph $D$ with $r\in V(D)$ is an $r$-flame if for every ${v\in V(D)-r}$, the in-degree of $v$ is equal to the local edge-connectivity $\lambda_D(r,v)$. We show that for every digraph $D$ and $r\in V(D)$, the edge sets of the $r$-flame subgraphs of $D$ form a greedoid. Our method yields a new proof of Lov\'asz' theorem stating: for every digraph $D$ and $r\in V(D)$, there is an $r$-flame subdigraph $F$ of $D$ such that $\lambda_F(r,v) =\lambda_D(r,v)$ for $v\in V(D)-r$. We also give a strongly polynomial algorithm to find such an $F$ working with a fractional generalization of Lov\'asz' theorem

Polymatroidal flows with lower bounds

AbstractThe polymatroidal network flow model is generalized to allow for supermodular lower bounds on flow in addition to submodular capacities. For arbitrary supermodular lower bounds and submodular capacities it is shown to be NP-hard simply to determine if a feasible integral flow exists. However, the situation is much more tractable if lower bound functions and capacity functions are in a relation of ‘compliance’. In this case, the Augmenting Path Theorem, the Integrality Theorem, the Max-Flow Min-Cut Theorem and the maximal flow algorithm of polymatroidal flows all have natural generalizations. A procedure for finding a feasible flow closely parallels one which is used for ordinary flow networks with lower bounds on arc flow