Given a stochastic matrix $A \in [0, 1]^{n \times n}$, we want to have a bound of its permanent $\operatorname{per}(A)$.
We do not want a general bound like that for the permanent of a doubly stochastic matrix which gives $\operatorname{per}(A) \geq n! / n^n$. Instead, we aim to find an instance-specific bound that can be computed efficiently. For example, we can easily derive that $\operatorname{per}(A) \geq 1 - \sum_{i \neq j} A_i \cdot A_j$, which gives a bound in polynomial time.
Can we have a tighter bound?