# Rank relation to maximum subpermanent and subdeterminant?

Given a $\pm1$ matrix $M$ of rank $r$ let the largest subdeterminant be $d$ and let the largest subpermanent be $p$.

1. Are there relations/bounds that connect $r$, $d$ and $p$?

2. Are there geometric and combinatorial interpretations of $d$ and $p$?

Have these notions been studied?