A double scaling algorithm for the constrained maximum flow problem

The constrained maximum flow problem is to send the maximum possible flow from a source node to a sink node in a directed capacitated network subject to a budget constraint that the total cost of the flow can be at most D  . In this research, we present a double scaling algorithm whose generic version runs in O(n2mlogmlogUlog(nC)) time, where n is the number of nodes in the network; m, the number of arcs; C, the largest arc cost; and U  , the largest arc capacity. This running time can be further reduced to O(n3logmlogUlog(nC)) with the wave implementation   of the cost scaling algorithm, and to O(nmlog(n2/m)logmlogUlog(nC))O(nmlog(n2/m)logmlogUlog(nC)) with the use of dynamic tree  s. These bounds are better than the current bound of O(mS(n,m,nC)logU), where S(n,m,nC)S(n,m,nC) is the time to find a shortest path from a single source to all other nodes where nonnegative reduced costs are used as arc costs.