A Dichotomy Theorem for Homomorphism Polynomials

TL;DR

This paper establishes a clear complexity dichotomy for homomorphism polynomials, showing they are either efficiently computable or VNP-complete, based on properties of the graph H.

Contribution

It proves a dichotomy theorem classifying the complexity of homomorphism polynomials for all graphs H, linking graph properties to computational complexity.

Findings

Polynomials for graphs with loops or no edges are computable in constant depth circuits.

Polynomials for other graphs are VNP-complete, indicating computational hardness.

Hardness results over the rationals for specific polynomials related to cut elimination.

Abstract

In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by arithmetic circuits in constant depth if H has a loop or no edge and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P). We also demonstrate the hardness over the rational field of cut eliminator, a polynomial defined by B\"urgisser which is known to be neither VP nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is the class of polynomials computable by arithmetic circuit of polynomial size).