Minimum weights needed to derandomize weight assignment by isolation lemma Under isolation lemma if you have a graph with $2n$ vertices and $m$ edges an isolating weight assignment can be obtained by assigning edges weights randomly from $\{1,2,\dots,2m-1,2m\}$. A weight assignment is isolating if there is an unique perfect matching with minimum sum weight.

Is $\Omega(m)$ weight necessary or would it be possible that there is a (deterministic) scheme that can get the job done in $O(\sqrt m)$?

Note that the best weights we know uses $O(2^m)$.
A deterministic scheme with tiny weights is easy to achieve: first, compute an arbitrary perfect matching, deterministically. Then, give the matched edges weight zero and all other edges weight one. The difficult question is how to compute a deterministic isolating scheme more quickly than finding a perfect matching.