A node-capacitated Okamura-Seymour theorem

作者：Lee James R^*; Mendel Manor; Moharrami Mohammad

来源：Mathematical Programming, 2015, 153(2): 381-415.

DOI：10.1007/s10107-014-0810-0

The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal epsilon > 0, if the node cut conditions are satisfied, then one can simultaneously route an epsilon-fraction of all the demands. This answers a question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of the multi-commodity polymatroid networks introduced by Chekuri et al. (ITCS, pp 399-408, 2012). In their framework, one associates to each node a submodular function on the adjacent edges that dictates the types of flows the node can support. In order to round the convex programs corresponding to node and polymatroid-capacitated flows, we devise a new type of random metric embedding that preserves some of the combinatorial structure of the underlying graph.

• 出版日期2015-11