Enumeration of the Monomials of a Polynomial and Related Complexity Classes

TL;DR

This paper explores efficient algorithms for enumerating monomials of polynomials, introduces new probabilistic complexity classes for enumeration, and demonstrates both algorithmic advancements and hardness results in polynomial monomial enumeration.

Contribution

It presents new algorithms with good delay for restricted polynomials, introduces probabilistic enumeration complexity classes, and establishes hardness results for degree 2 polynomial monomial enumeration.

Findings

New algorithms with good delay for restricted polynomials

Introduction of TotalPP, IncPP, DelayPP classes for enumeration complexity

Hardness of finding specific monomials in degree 2 polynomials

Abstract

We study the problem of generating monomials of a polynomial in the context of enumeration complexity. In this setting, the complexity measure is the delay between two solutions and the total time. We present two new algorithms for restricted classes of polynomials, which have a good delay and the same global running time as the classical ones. Moreover they are simple to describe, use little evaluation points and one of them is parallelizable. We introduce three new complexity classes, TotalPP, IncPP and DelayPP, which are probabilistic counterparts of the most common classes for enumeration problems, hoping that randomization will be a tool as strong for enumeration as it is for decision. Our interpolation algorithms proves that a lot of interesting problems are in these classes like the enumeration of the spanning hypertrees of a 3-uniform hypergraph. Finally we give a method to interpolate a degree 2 polynomials with an acceptable (incremental) delay. We also prove that finding a specified monomial in a degree 2 polynomial is hard unless RP = NP. It suggests that there is no algorithm with a delay as good (polynomial) as the one we achieve for multilinear polynomials.