Computing Gaussian and exponential integrals in ${\mathbb{R}}^n$

We consider expectations of the type $E\exp \left\{{\sum }_{i=1}^m{\phi }_i\right\}$, where ${\phi }_i:{\mathbb{R}}^n⟶\mathbb{C}$ are functions, each depending on a few coordinates of a point in ${\mathbb{R}}^n$, and the expectation is taken with respect to the standard Gaussian or symmetric exponential probability measures. We prove sufficient conditions, in terms of the Lipschitz constants of ${\phi }_i$ and the combinatorics of their dependencies, for the integral to be non-zero, and, consequently, to be amenable to a computationally efficient approximation. We discuss applications to computing volumes of bodies and statistics on integer points in polyhedra in ${\mathbb{R}}^n$.