Before revision:  Our improvement is achieved by introducing a new variant of the push-relabel algorithm for computing maximum flows. We use our improved algorithm for maximum flow in  k-apex graphs , together with additional insights, to obtain an O(k^3 n \\textrm{polylog (nC))-time algorithm for computing maximum integer flows in planar graphs with integer arc and vertex capacities bounded by C, and k sources and sinks.
After revision:  Our improvement is achieved by introducing a new variant of the push-relabel algorithm of Goldberg and Tarjan and use it to find a maximum flow in k-apex graphs , together with additional insights, to obtain an O(k^3 n \\textrm{polylog (nC))-time algorithm for computing maximum integer flows in planar graphs with integer arc and vertex capacities bounded by C, and k sources and sinks.