Parallel Polynomial Permanent Mod Powers of 2 and Shortest Disjoint Cycles

TL;DR

This paper introduces a parallel algorithm for computing the permanent modulo 2^k for matrices of univariate integer polynomials, placing the problem in a low-depth complexity class and improving algorithms for shortest disjoint paths and cycles.

Contribution

It extends existing techniques to develop a parallel algorithm for permanent mod 2^k and shortest disjoint cycles, advancing parallel approaches for these problems.

Findings

Permanent mod 2^k is in ParityL subset of NC^2.

New randomized parallel algorithms for shortest 2-disjoint paths.

Algorithms for shortest cycles passing through fixed vertices or edges.

Abstract

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.