Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes

We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2018). We consider three different variants of graph homomorphisms, namely injective homomorphisms, directed homomorphisms, and injective directed homomorphisms, and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties: The polynomial families complete for VF, VBP, and VP are model independent, i.e., they do not use a particular instance of a formula, algebraic branching programs, or circuit for characterising VF, VBP, or VP, respectively. All the polynomial families are hard under p-projections.