I wish to find an algorithm that reduces a matrix over $\mathbb{Z}_2$ to a permutation matrix using as few row summing operations as possible.
Standard Gaussian elimination turns a matrix into the identity matrix and requires $O(n^2/\log n)$ row sums (https://dl.acm.org/citation.cfm?id=2011767). Is there a way to achieve better results when you only care about reducing to a permutation? Asymptotic bounds would be nice, but I'm primarily interested in heuristic performance.