An FPTAS for minimizing a class of low-rank quasi-concave functions over a convex set

We consider the problem of minimizing a class of quasi-concave functions over a convex set. Quasiconcave functions are a generalization of concave functions and thus, NP-hard to minimize over a convex set in general. We present a simple fully polynomial time approximation scheme (FPTAS) for minimizing a fairly general class of low-rank quasi-concave functions. Our algorithm is based on solving a polynomial number of linear minimization problems over a convex set and computes a near-optimal solution that is an extreme point of the convex set. Therefore, it applies directly to combinatorial 0-1 problems for which the convex hull of feasible solutions is known, such as shortest paths, spanning trees and matchings in undirected graphs.