Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes

Abstract

We present polynomial families complete for the well-studied algebraic complexity classes \(\textsf {VF}\), \(\textsf {VBP}\), \(\textsf {VP}\), and \(\textsf {VNP}\). The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. [10]. We consider three different variants of graph homomorphisms, namely injective homomorphisms, directed homomorphisms and injective directed homomorphisms and obtain polynomial families complete for \(\textsf {VF}\), \(\textsf {VBP}\), \(\textsf {VP}\), and \(\textsf {VNP}\) under each one of these. The polynomial families have the following properties: The polynomial families complete for \(\textsf {VF}\), \(\textsf {VBP}\), and \(\textsf {VP}\) are model independent, i.e. they do not use a particular instance of a formula, ABP or circuit for characterising \(\textsf {VF}\), \(\textsf {VBP}\), or \(\textsf {VP}\), respectively. All the polynomial families are hard under p-projections.