We present a parallel algorithm for permanent mod 2^k of a matrix of univariate integer polynomials. It places the problem in ParityL subset of NC^2. This extends the techniques of [Valiant], [Braverman, Kulkarni, Roy] and [Bj\"orklund, Husfeldt], and yields a (randomized) parallel algorithm for shortest 2-disjoint paths improving upon the recent result from (randomized) polynomial time. We also recognize the disjoint paths problem as a special case of finding disjoint cycles, and present (randomized) parallel algorithms for finding a shortest cycle and shortest 2-disjoint cycles passing through any given fixed number of vertices or edges.

中文翻译：

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2 和最短不相交循环的平行多项式永久模幂

我们提出了一种用于单变量整数多项式矩阵的永久模 2^k 的并行算法。它将问题放在 NC^2 的 ParityL 子集中。这扩展了 [Valiant]、[Braverman、Kulkarni、Roy] 和 [Bj\"orklund, Husfeldt] 的技术，并产生了一种用于最短 2 不相交路径的（随机）并行算法，改进了（随机）多项式的最近结果我们还将不相交路径问题视为寻找不相交循环的特殊情况，并提出了（随机化的）并行算法，用于寻找最短循环和最短的 2 个不相交循环通过任何给定的固定数量的顶点或边。