Lower Bounds in Algebraic Complexity via Symmetry and Homomorphism Polynomials
Prateek Dwivedi, Benedikt Pago, Tim Seppelt
TL;DR
This work develops a symmetric analogue of Valiant’s algebraic complexity framework, defining the classes $\mathsf{symVF}$, $\mathsf{symVBP}$, and $\mathsf{symVP}$ as matrix-symmetric counterparts to $\mathsf{VF}$, $\mathsf{VBP}$, and $\mathsf{VP}$. The authors establish unconditional separations $\mathsf{symVF} \subsetneq \mathsf{symVBP} \subsetneq \mathsf{symVP}$ by characterising these classes via linear combinations of homomorphism polynomials for patterns of bounded treedepth, pathwidth, and treewidth, respectively. They further show that symmetric classes contain $\mathsf{VBP}$- and $\VP$-complete polynomials and provide general graph-theoretic criteria for hardness of homomorphism polynomials, strengthening lower bounds in the non-symmetric setting. A key insight is the link between circuit orbit size, gate support depth, and pattern treedepth, which yields a semantic separation (counting width) that goes beyond simple width-based arguments. Conditional results connect symmetry to parametrised complexity, notably via $\mathsf{VFPT} \neq \mathsf{VW}[1]$, revealing a deep connection between symmetric and non-symmetric computation for homomorphism polynomials and suggesting broader implications for Valiant’s program.
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.