Abstract
The max-flow problem in planar networks with only edge capacities has been proved to be in NC (Nickle's Class). This paper considers a more general version of the problem when the nodes as well as the edges have capacities. In a general network, the node-edge-capacity problem can be easily reduced to the edge-capacity problem. But in the case of planar network this reduction may destroy the planarity, and reduces the problem to the edge-capacity problem in a general network, which is P-complete. A recent contribution presents a new reduction for planar networks, that maintains the planarity. In this paper, it is proved that this reduction is in NC and thus the node-edge-capacity problem in undirected planar networks is in NC.
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Supported by the National Basic Research 973 Program of China under Grant No.G1999032700.
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Zhang, XC., Wan, YY. & Chen, GL. Max-flow problem in undirected planar networks with node capacities being in NC. J. Comput. Sci. & Technol. 19, 787–790 (2004). https://doi.org/10.1007/BF02973440
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DOI: https://doi.org/10.1007/BF02973440