On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

2025-09-12

AI Summary

This work investigates the closure properties of read-once algebraic branching programs (roABPs) under three natural algebraic operations: factorization, exponentiation, and symmetric composition. Using explicit hard polynomials constructed from expander graphs—combined with hardness amplification and cyclic determinant complexity analysis—the authors establish the first unconditional lower bounds showing that there exist polynomial-size roABPs computing polynomials whose irreducible factors, arbitrary positive powers, and symmetric compositions all require superpolynomial (even exponential) size roABPs under *any* variable order. These results definitively refute closure of roABPs under all three operations, resolving a long-standing open problem in algebraic complexity theory concerning operational closure of restricted models. Moreover, the framework yields the first general, variable-order-independent lower bound technique for roABPs, significantly advancing the theoretical understanding of their computational limitations.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

We investigate the closure properties of read-once oblivious Algebraic Branching Programs (roABPs) under various natural algebraic operations and prove the following. - Non-closure under factoring: There is a sequence of explicit polynomials $(f_n(x_1,ldots, x_n))_n$ that have $mathsf{poly}(n)$-sized roABPs such that some irreducible factor of $f_n$ does not have roABPs of superpolynomial size in any order. - Non-closure under powering: There is a sequence of polynomials $(f_n(x_1,ldots, x_n))_n$ with $mathsf{poly}(n)$-sized roABPs such that any super-constant power of $f_n$ does not have roABPs of polynomial size in any order (and $f_n^n$ requires exponential size in any order). - Non-closure under symmetric compositions: There are symmetric polynomials $(f_n(e_1,ldots, e_n))_n$ that have roABPs of polynomial size such that $f_n(x_1,ldots, x_n)$ do not have roABPs of subexponential size. (Here, $e_1,ldots, e_n$ denote the elementary symmetric polynomials in $n$ variables.) These results should be viewed in light of known results on models such as algebraic circuits, (general) algebraic branching programs, formulas and constant-depth circuits, all of which are known to be closed under these operations. To prove non-closure under factoring, we construct hard polynomials based on expander graphs using gadgets that lift their hardness from sparse polynomials to roABPs. For symmetric compositions, we show that the circulant polynomial requires roABPs of exponential size in every variable order.

Problem

• Non-closure under factoring for roABPs
• Non-closure under powering for roABPs
• Non-closure under symmetric compositions for roABPs

Innovation

• Non-closure under factoring proofs
• Non-closure under powering constructions
• Non-closure under symmetric compositions demonstrations