  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.
