On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

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.