The Complexity of Order-Finding for ROABPs

We study the \emph{order-finding problem} for Read-once Oblivious Algebraic Branching Programs (ROABPs). Given a polynomial $f$ and a parameter $w$, the goal is to find an order $\sigma$ in which $f$ has an ROABP of \emph{width} $w$. We show that this problem is NP-hard in the worst case, even when the input is a constant degree polynomial that is given in its dense representation. We provide a reduction from CutWidth to prove these results. Owing to the exactness of our reduction, all the known results for the hardness of approximation of Cutwidth also transfer directly to the order-finding problem. Additionally, we also show that any constant-approximation algorithm for the order-finding problem would imply a polynomial time approximation scheme (PTAS) for it. On the algorithmic front, we design algorithms that solve the order-finding problem for generic ROABPs in polynomial time, when the width $w$ is polynomial in the individual degree $d$ of the polynomial $f$. That is, our algorithm is efficient for most/random ROABPs, and requires more time only on a lower-dimensional subspace (or subvariety) of ROABPs. Even when the individual degree is constant, our algorithm runs in time $n^{O(\log w)}$ for most/random ROABPs. This stands in strong contrast to the case of (Boolean) ROBPs, where only heuristic order-finding algorithms are known.