On the Hardness and Smoothed Complexity of Quasi-Concave Minimization

In this paper, we resolve, the smoothed and approximative complexity of low-rank quasi-concave minimization, providing both upper and lower bounds. As an upper bound, we provide the first smoothed analysis of quasi-concave, minimization. The analysis is based on a smoothed bound for the number of extreme points of the projection of the feasible polytope onto a k-dimensional subspace. where k is the rank (informally, the dimension of nonconvexity)ofthe quasi-concave function. Our smoothed bound is polynomial in the original dimension of the problem n and the perturbation size p. and it is exponential in the rank of the function k. From this, we obtain the first randomized fully polynomial-time approximation scheme for low-rank quasi-concave minimization under broad conditions. In contrast with this, we prove log n-hardness of approximation for general quasi-concave minimization. This shows that our smoothed bound is essentially tight, in that no polynomial smoothed bound is possible for quasi-concave functions of general rank k. The tools that we introduce for the smoothed analysis may be of independent interest. All previous smoothed analyses of polytopes analyzed projections onto two-dimensional subspaces and studied them using trigonometry to examine the angles between vectors and 2-planes in Ropf". In this paper, we provide what is, to our knowledge, the first smoothed analysis of the projection of polytopes onto higher-dimensional subspaces. To do this, we replace the trigonometry with tools from random matrix theory and differential geometry on the Grassmannian. Our hardness reduction is based on entirely different proofs that may also be of independent interest; we show that the stochastic 2-stage minimum spanning tree problem has a supermodular objective and that supermodular minimization is hard to approximate.