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Some new results on node-capacitated packing of A-paths

In this paper we propose a (semi-strongly) polynomial time algorithm to find a maximum packing subject to node-capacities, and thus we obtain a generalization of Keijsper, Pendavingh and Stougie algorithm concerning edge-capacities. Our method is based on Gerards' strongly polynomial time algorithm to find a maximum b-matching in a graph, which is based on a so-called Proximity Lemma. Our node-capacitated A-path packing algorithm first constructs a maximum fractional packing by using an ellipsoid method subroutine, then takes its integer part to obtain a near-optimal integral packing, and finally we construct a maximum integer packing by a short sequence of augmentations. This short sequence of augmentations is constructed by applying the version of Gerards' Proximity Lemma, specially formulated for the node-capacitated A-path packing problem.
 In addition, we also state some related results on the fractional packing problem. We prove the primal- and dual integrality of the corresponding linear program. We mention that the fractional packing problem reduces to the matroid fractional matching problem.