Volume of polyhedron

When $\boldsymbol{r}=\boldsymbol{c}=(1,1,\dots,1)$ the polytope $X$ is the Birkhoff polytope of $n\times n$ doubly-stochastic matrices. Computing its $(n-1)^2$-dimensional volume is a well-known difficult problem. An answer is given by De Loera, Liu, and Yoshida in http://arxiv.org/abs/math/0701866, but it is quite a complicated formula. For generalizing it to $\boldsymbol{r}$ and $\boldsymbol{c}$ and to higher dimension, see De Loera's survey at http://arxiv.org/pdf/1307.0124.pdf. For additional information in the case of the Birkhoff polytope, see OEIS A078524, A078525, A037302.