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.

翻译：我们研究了只读一次遗忘代数分支程序（ROABP）的排序问题。给定一个多项式$f$和参数$w$，该问题的目标是找到一个顺序$\sigma$，使得$f$在该顺序下存在宽度为$w$的ROABP。我们证明在最坏情况下，即使输入是稠密表示形式的常次数多项式，该问题也是NP难的。我们通过从割宽度问题的归约来证明这些结果。由于归约的精确性，所有已知的关于割宽度近似难度的结论都可以直接迁移到排序问题上。此外，我们还证明任何针对排序问题的常数近似算法都将意味着该问题存在多项式时间近似方案（PTAS）。在算法方面，我们设计了多项式时间内求解通用ROABP排序问题的算法，前提是宽度$w$相对于多项式$f$的个体次数$d$是多项式规模的。也就是说，我们的算法对大多数/随机ROABP是高效的，仅在ROABP的低维子空间（或子簇）上需要更多时间。即使当个体次数为常数时，我们的算法对大多数/随机ROABP也仅需$n^{O(\log w)}$时间。这与（布尔型）ROBP的情形形成鲜明对比——后者目前仅存在启发式排序算法。