Abstract
We present a polynomial-time algorithm for finding a minimum weight feedback arc set (FAS) in arc-weighted reducible flow graphs and for finding a minimum weight feedback vertex set (FVS) in vertex-weighted reducible flow graphs. The algorithm has time complexity O(mn 2 log( n 2 m )) , where n is the number of vertices and m is the number of arcs in the reducible flow graph. For unweighted reducible flow graphs, the algorithm for FAS has time complexity O( min(mn 5 3 , m 2)) . We also show that any algorithm that solves the FAS problem or the vertex-weighted FVS problem on reducible flow graphs has time complexity at least that of finding a minimum cut in a flow network, for which the best algorithm currently known has time complexity Θ(mn log( n 2 m )) for networks with arbitrary capacities, and Θ( min(mn 2 3 , m 3 2 )) for networks with unit capacities. Our results establish several connections between the FAS problem on reducible flow graphs and the minimum cut-maximum flow problem on flow networks.
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