On the complexity and approximability of budget-constrained minimum cost flows

We investigate the complexity and approximability of the budget-constrained minimum cost flow problem, which is an extension of the traditional minimum cost flow problem by a second kind of costs associated with each arc, whose total value in a feasible flow is constrained by a given budget B. This problem appears both in practical applications and as a subproblem when applying the ε-constraint method to the bicriteria minimum cost flow problem. We show that we can solve the problem exactly in weakly polynomial time O(log⁡M⋅MCF(m,n,C,U)), where C, U, and M are upper bounds on the largest absolute cost, largest capacity, and largest absolute value of any number occurring in the input, respectively, and MCF(m,n,C,U) denotes the complexity of finding a traditional minimum cost flow. Moreover, we present two fully polynomial-time approximation schemes for the problem on general graphs and one with an improved running-time for the problem on acyclic graphs.