The evaluation of specific integrals over moduli spaces of metric graphs is a central bottleneck in extracting predictions from quantum field theory (QFT) in physics. I will begin with a gentle introduction to these spaces, the combinatorics of the underlying objects, and the

physical integrals defined upon them.

A central role in this framework is played by the relevant matroidal structures of (Feynman) graphs and the properties of associated geometries, such as generalized permutahedra.

Drawing an analogy to Maryam Mirzakhani's recursive solution for Weil-Petersson volumes of the moduli space of curves, I explain how a tropical deformation of the quantum field theory problem yields volume forms that can be integrated via combinatorial recursions. This

framework, for instance, leads to an efficient, polynomial-time, sampling algorithm for metric graphs approximately weighted by their QFT

contributions, which in turn provides a new faster method for QFT computations.