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) of the quasi-concavefunction. Our smoothed bound is polynomial in the original dimension of the problem n and the perturbation size ρ, and it is exponential in the rank of the function k. From this, we obtain the first randomized fully polynomialtime 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 f...

• Type: Conference
• Year: 2007
• Where: FOCS
• Authors: Jonathan A. Kelner, Evdokia Nikolova
• Keywords: FOCS 2007, Smoothed, Smoothed Analysis, Smoothed Bound, Theoretical Computer Science