The next generations of mobile networks will be deployed as ultra-dense networks, to match the demand for increased capacity and the challenges that communications in the higher portion of the spectrum (i.e., the mmWave band) introduce. Ultra-dense networks, however, require pervasive, high-capacity backhaul solutions, and deploying fiber optic to all base stations is generally considered to be too expensive for network operators. The 3rd Generation Partnership Project (3GPP) has thus introduced Integrated Access and Backhaul (IAB), a wireless backhaul solution in which the access and backhaul links share the same hardware, protocol stack, and also spectrum. The multiplexing of different links in the same frequency bands, however, introduces interference and capacity sharing issues, thus calling for the introduction of advanced scheduling and coordination schemes. This paper proposes a semi-centralized resource allocation scheme for IAB networks, designed to be flexible, with low complexity, and compliant with the 3GPP IAB specifications. We develop a version of the Maximum Weighted Matching (MWM) problem that can be applied on a spanning tree that represents the IAB network and whose complexity is linear in the number of IAB-nodes. The proposed solution is compared with state-of-the-art distributed approaches through end-to-end, full-stack system-level simulations with a 3GPP-compliant channel model, protocol stack, and a diverse set of user applications. The results show how that our scheme can increase the throughput of cell-edge users up to 5 times, while decreasing the overall network congestion with an end-to-end delay reduction of up to 25 times.