A note on VNP-completeness and border complexity

TL;DR

This paper investigates the topological structure of VNP-complete polynomials, showing that their set and its complement are dense in VNP under various reduction notions, revealing surprising properties of algebraic complexity classes.

Contribution

It provides the first structured study of border reduction notions and demonstrates the density of VNP-complete polynomials and their complement within VNP.

Findings

VNP  VNPC is dense in VNP

VNPC is dense in VNP

Density holds under multiple reduction notions

Abstract

In 1979 Valiant introduced the complexity class VNP of p-definable families of polynomials, he defined the reduction notion known as p-projection and he proved that the permanent polynomial and the Hamiltonian cycle polynomial are VNP-complete under p-projections. In 2001 Mulmuley and Sohoni (and independently B\"urgisser) introduced the notion of border complexity to the study of the algebraic complexity of polynomials. In this algebraic machine model, instead of insisting on exact computation, approximations are allowed. This gives VNP the structure of a topological space. In this short note we study the set VNPC of VNP-complete polynomials. We show that the complement VNP \ VNPC lies dense in VNP. Quite surprisingly, we also prove that VNPC lies dense in VNP. We prove analogous statements for the complexity classes VF, VBP, and VP. The density of VNP \ VNPC holds for several different reduction notions: p-projections, border p-projections, c-reductions, and border c-reductions. We compare the relationship of the completeness notions under these reductions and separate most of the corresponding sets. Border reduction notions were introduced by Bringmann, Ikenmeyer, and Zuiddam (JACM 2018). Our paper is the first structured study of border reduction notions.