We study the problem of minimizing a nonnegative separable concave function over a compact feasible set. We approximate this problem to within a factor of 1 + ǫ by a piecewise-linear minimization problem over the same feasible set. Our main result is that when the feasible set is a polyhedron, the number of resulting pieces is polynomial in the input size of the polyhedron and linear in 1/ǫ. For many practical concave cost problems, the resulting piecewise-linear cost problem can be formulated as a wellstudied discrete optimization problem. As a result, a variety of polynomial-time exact algorithms, approximation algorithms, and polynomial-time heuristics for discrete optimization problems immediately yield fully polynomial-time approximation schemes, approximation algorithms, and polynomial-time heuristics for the corresponding concave cost problems.We illustrate our approach on two problems. For the concave cost multicommodity flow problem, we devise a new heuristic and study its performance using computational experiments. We are able to approximately solve significantly larger test instances than previously possible, and obtain solutions on average within 4.27% of optimality. For the concave cost facility location problem, we obtain a new 1.4991 + ǫ approximation algorithm. * This research is based on the second author's Ph.D. thesis at the Massachusetts Institute of Technology [Str08]. An extended abstract of this research has appeared in [MS04].

We introduce an algorithm design technique for a class of combinatorial optimization problems with concave costs. This technique yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the fixedcharge counterpart of the problem. For many practical concave cost problems, the fixed-charge counterpart is a well-studied combinatorial optimization problem. Our technique preserves constant factor approximation ratios, as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms.Using our technique, we obtain a new 1.61-approximation algorithm for the concave cost facility location problem. For inventory problems, we obtain a new exact algorithm for the economic lot-sizing problem with general concave ordering costs, and a 4-approximation algorithm for the joint replenishment problem with general concave individual ordering costs.

We study the minimum-cost flow problem on dynamic networks with nonlinear cost functions that depend on time and edge flow. A general procedure for solving the problem using the time-expanded network is described. The main properties of dynamic flows on networks with concave cost functions are studied. We propose an algorithm for finding the optimal flow in networks with exactly one source; its running time is polynomial for a fixed number of sinks. Some details concerning the correctness of the algorithm and its computational complexity are discussed.

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