Minimal-cost network flow problems with variable lower bounds on arc flows
Computers and Operations Research
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We consider a variant of the well-known minimum cost flow problem where the flow on each arc in the network is restricted to be either zero or above a given lower bound. The problem was recently shown to be weakly NP-complete even on series-parallel graphs. We start by showing that the problem is strongly NP-complete and cannot be approximated in polynomial time (unless P=NP) up to any polynomially computable function even when the graph is bipartite and the given instance is guaranteed to admit a feasible solution. Moreover, we present a pseudo-polynomial-time exact algorithm and a fully polynomial-time approximation scheme (FPTAS) for the problem on series-parallel graphs.