Maximizing Non-Linear Concave Functions in

TLDR: Using this method, the first polynomial approximation algorithms for many NP-hard problems such as the parametric Euclidean Traveling Salesman Problem are obtained.

Abstract: Consider a convex set P in Rd and a piecewise polynomial concave function F: P -+ R. Let A be an algorithm that given a point x E Rd computes F(x) if x E PI or returns a concave polynomial p such that p(x) < 0 but for any y E P, p(y) 2 0. We assume that d is fixed and that all comparisons in A depend on the sign of polynomial functions of the input point. We show that under these conditions, one can find maxp F in time which is polynomial in the number of arithmetic operations of A. Using our method we give the first strongly polynomial algorithms for many nonlinear parametric problems in fixed dimension, such as the parametric mar pow problem, the parametric minimum s-t distance, the parametric spanning tree problem and other problems. In addition we show that in one dimension, the same result holds even if we only know how to approximate the value of F. Specifically, if we can obtain an a-approximation for F(x) then we can a-approximate the value of maxF. We thus obtain the first polynomial approximation algorithms for many NP-hard problems such as the parametric Euclidean Traveling Salesman Problem.