Lower Bounds in Algebraic Complexity via Symmetry and Homomorphism Polynomials

2026-01-14

AI Summary

This work addresses the challenge of establishing unconditional lower bounds in algebraic complexity theory by focusing on symmetric computation models. We introduce the symmetric algebraic complexity classes symVP, symVBP, and symVF, characterizing their computational power via orbit-size polynomials. For the first time, we unconditionally prove the strict inclusions symVF ⊊ symVBP ⊊ symVP and establish connections between these classes and graph homomorphism polynomials whose underlying graphs have bounded treedepth or pathwidth. Furthermore, under the parameterized complexity assumption VFPT ≠ VW[1], we demonstrate the completeness of several homomorphism polynomials for the classes VBP, VP, and VNP, thereby significantly expanding the known landscape of natural complete problems in algebraic complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Applications of cryptography

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.

Problem

• symmetric algebraic complexity
• homomorphism polynomials
• complexity classes
• lower bounds
• Valiant's conjecture

Innovation

• symmetric algebraic circuits
• homomorphism polynomials
• algebraic complexity classes
• treewidth
• unconditional lower bounds