I have a question regarding efficient and possibly simple algorithms for computing volumes of $n$-dimensional polytopes.
The polytope of concern isn't arbitrary: it is obtained by applying a linear transformation to a unit hypercube, whose associated matrix has the form $I - L$, where $I$ is identity matrix and $L$ is an arbitrary (in reality sparse) matrix with zeros on its diagonal.
The main difficulty of the problem is that I do not need to calculate the whole volume of the polytope, but only the volume of the part lying in the first hyperoctant (i.e. whose point have all the coordinates positive). So what I really need is an algorithm to compute quickly and efficiently the volume of the intersection of a polytope and the first hyperoctant.
If it helps to know the context of the question, I am doing research in the field of complex networks, where $L$ is the adjacency matrix of a $E-R$ network. The hypervolume which I am concerned with would be a sort of a measure of resiliency of the network.