Reconstructing the as-built architectural shape of building interiors has emerged in recent years as an important and challenging research problem. An effective approach must be able to faithfully capture the architectural structures and separate permanent components from clutter (e.g. furniture), while at the same time dealing with defects in the input data. For many applications, higher-level information on the environment is also required, in particular the shape of individual rooms. To solve this ill-posed problem, state-of-the-art methods assume constrained input environments with a 2.5D or, more restrictively, a Manhattan-world structure, which significantly restricts their applicability in real-world settings. We presents a novel pipeline that allows to reconstruct general 3D interior architectures, significantly increasing the range of real-world architectures that can be reconstructed and labeled by any interior reconstruction method to date. Our method finds candidate permanent components by reasoning on a graph-based scene representation, then uses them to build a 3D linear cell complex that is partitioned into separate rooms through a multi-label energy minimization formulation. We demonstrate the effectiveness of our method by applying it to a variety of real-world and synthetic datasets and by comparing it to more specialized state-of-the-art approaches.