@Dustin G. Mixon, do you think it is possible to set up flows and capacities in such way that F-F algorithm becomes usable ?

@David Benson-Putnins, I mentioned the boundary as an illustration. Obviously the 0s inside the $G'$ need to be considered as well.

@Dustin G. Mixon, you are right, the problem did not say it clearly that $G'$ has all 0 and my previous comment is wrong. $G-G'$ contains only {1,-1}.

@David Benson-Putnins, My apologies for possible misrepresentation. Set $G'$ indeed contains only vertices with values 0, however, intuitively speaking, the vertices that are close to the boundary have edges that reach out to the $G-G'$ where vertices can have labels from {0,-1,1}. So yes, vertices in $G-G'$ can have any value(either 0, 1 or -1)

@Dustin G. Mixon, thank you for your input. Although on the surface the max-cut seems to be similar, in reality the problems are quite different. The challenge in the problem I posted is to chose values of nodes on any given edge such that the product of such values $l_u l_v$ stays non-negative. Since we have boundary between $G$ and $G'$ such that edges intersected by the boundary on the side of $G'$ have nodes with values 0 and on the side of $G$ with have values{0,-1,1} the challenge is to relabel the 0s on the side of $G'$ in such way that product of labels on both sides maximizes the sum