Working Paper MASSACHUSETTS INSTITUTE OF TECHNOLOGY by Separable Concave Optimization Approximately Equals Piecewise Linear Optimization

Consider a separable concave minimization problem with nondecreasing costs over a general ground set X ⊆ R+. We show how to efficiently approximate this problem to a factor of 1+2 in optimal cost by a single piecewise linear minimization problem over X. The number of pieces is linear in 1/2 and polynomial in the logarithm of certain ground set parameters; in particular, it is independent of the cost functions. Our main result is that when the minimization is over a polyhedron, the number of pieces, and thus the size of the resulting problem, is polynomial in the input size of the polyhedron and linear in 1/2. We present generalizations to problems with grounds sets not contained in R+ and concave functions that are not monotone. Our approach provides a general technique for applying discrete optimization methods to practical concave cost problems with polyhedral ground sets. We exemplify the approach on two problems. For the concave cost multicommodity flow problem, we devise an approximate computational solution procedure using our technique and a primal-dual solution procedure. We are able to solve randomly generated instances significantly larger than previously possible, and obtain solutions within 4% of optimality on average. For the lot-sizing problem with concave production costs, we derive an algorithm with a new polynomial running time that is not dominated by that of previously known algorithms.