Lower Bounds in Algebraic Complexity via Symmetry and Homomorphism Polynomials

TL;DR

This paper develops a symmetric algebraic complexity theory, defining classes like symVP, symVBP, and symVF, and proves their strict inclusions, characterizing polynomials via homomorphism polynomials and establishing new lower bounds.

Contribution

It introduces symmetric analogues of VP, VBP, and VF, and provides unconditional separations and characterizations using homomorphism polynomials with bounded treedepth and pathwidth.

Findings

Unconditional strict inclusions: symVF ⊂ symVBP ⊂ symVP.

Characterization of symVF and symVBP via homomorphism polynomials.

Identification of homomorphism polynomials that are VBP-, VP-, or VNP-complete.

Abstract

Valiant's conjecture asserts that the circuit complexity classes VP and VNP are distinct, meaning that the permanent does not admit polynomial-size algebraic circuits. As it is the case in many branches of complexity theory, the unconditional separation of these complexity classes seems elusive. In stark contrast, the symmetric analogue of Valiant's conjecture has been proven by Dawar and Wilsenach (2020): the permanent does not admit symmetric algebraic circuits of polynomial size, while the determinant does. Symmetric algebraic circuits are both a powerful computational model and amenable to proving unconditional lower bounds. In this paper, we develop a symmetric algebraic complexity theory by introducing symmetric analogues of the complexity classes VP, VBP, and VF called symVP, symVBP, and symVF. They comprise polynomials that admit symmetric algebraic circuits, skew circuits, and formulas, respectively, of polynomial orbit size. Having defined these classes, we show unconditionally that $\mathsf{symVF} \subsetneq \mathsf{symVBP} \subsetneq \mathsf{symVP}$. To that end, we characterise the polynomials in symVF and symVBP as those that can be written as linear combinations of homomorphism polynomials for patterns of bounded treedepth and pathwidth, respectively. This extends a previous characterisation by Dawar, Pago, and Seppelt (2026) of symVP. Finally, we show that symVBP and symVP contain homomorphism polynomials which are VBP- and VP-complete, respectively. We give general graph-theoretic criteria for homomorphism polynomials and their linear combinations to be VBP-, VP-, or VNP-complete. These conditional lower bounds drastically enlarge the realm of natural polynomials known to be complete for VNP, VP, or VBP. Under the assumption VFPT $\neq$ VW[1], we precisely identify the homomorphism polynomials that lie in VP as those whose patterns have bounded treewidth.