# Flipping one bit to maximize BMM output

Consider a boolean matrix $$A$$ of size $$N \times N$$ and let $$A^\top$$ be its transpose. Let $$C = AA^\top$$ be the boolean matrix multiplication (BMM) result and let $$c$$ be the number of non-negative entries in $$C$$. My goal is to flip one bit in $$A$$ from $$0$$ to $$1$$ to create a new matrix $$B$$ such that $$C' = B B^\top$$ is the BMM result and if $$c'$$ is the number of non-negative entries in $$C'$$, we wish to maximize $$c' - c$$. In other words, by flipping one entry, we wish to $$\it maximize$$ the incremental output generated by BMM of the resulting matrix. Has this problem been studied in literature and if yes, is this problem as hard as BMM itself?

I am also interested in the decremental analogue of the problem where we can flip one bit in A from $$1$$ to $$0$$ to maximize the decrease in the output between BMM of the original matrix with its transpose and the new matrix with its transpose.