Optimization and NP_R-Completeness of Certain Fewnomials

TL;DR

This paper presents a polynomial-time approximation scheme for optimizing certain sparse multivariate functions, extending A-discriminant theory and identifying new NP_R-complete problems.

Contribution

It introduces a high-precision approximation method for honest n-variate (n+2)-nomials with real exponents, improving complexity bounds from exponential to polynomial.

Findings

Efficient approximation scheme for specific sparse polynomials

Extension of A-discriminant theory to real exponents

Identification of new NP_R-complete problems

Abstract

We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n+2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality checks, and are polynomial in n and the logarithm of a certain condition number. For the special case of polynomials (i.e., integer exponents), the log of our condition number is quadratic in the sparse encoding. The best previous complexity bounds were exponential in the sparse encoding, even for n fixed. Along the way, we extend the theory of A-discriminants to real exponents and certain exponential sums, and find new and natural NP_R-complete problems.