# Characterization for total dual integrality

A problem I study reduces to whether the polyhedron $$P=\{\mathbf{x}\mid A\mathbf{x}=\mathbf{1}, \mathbf{x}\geq0\}$$ is integral ($$A$$ is a matrix with coefficients in $$\{0,1\}$$). I know that the unimodularity of $$A$$ is a sufficient condition, which has been useful for me. I want to know whether there is any weaker condition.

I read from a textbook that $$P$$ is integral if the system $$A\mathbf{x}=\mathbf{1}, \mathbf{x}\geq0$$ is totally dual integral (TDI), i.e., the minimum in

$$\max\{\mathbf{c}^T\mathbf{x}\mid A\mathbf{x}=\mathbf{1}, \mathbf{x}\geq0\}=\min\{\mathbf{y}^T\cdot\mathbf{1}\mid \mathbf{y}^TA\geq \mathbf{c}^T\}$$

has an integral optimum solution $$\mathbf{y}$$ for each integral $$\mathbf{c}$$ for which the minimum is finite.

This definition is too abstract for me. I wonder whether there is any characterization for this condition that I can use directly.