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admissible cells algorithm arc capacities arc flows arc-chain assignment problem assumed assumption breakthrough capacity function column sums combinatorial computation constraints construction Corollary corresponding cost flow problem D. R. Fulkerson defined directed arcs directed cycle directed graph disconnecting set dual variables equations example existence feasible circulation feasible flow Figure flow augmenting path flow function follows given graph G hence Hitchcock problem incidence matrix integral kilter number L. R. Ford labeling process Lemma linear programming lower bounds Math max-flow min-cut theorem maximal dynamic flow maximal flow problem maximal flow value method minimal cost flow minimal cut necessary and sufficient network flows node-arc non-breakthrough non-negative optimality properties partially ordered set primal proof of Theorem reverse arcs Routine satisfying sequence set of arcs set of nodes shortest chain shown in Fig solving spanning subtree static flow subsets supply-demand theorem suppose Theorem 1.1 tree tt(x undirected unlabeled upper bounds zero