This paper studies the sum-capacity of the Multiple Input Single Output (MISO) Gaussian broadcast channel where K single-antenna users are served by a base station with N antennas, with N < K. The generalized Degrees-of-Freedom (gDoF) for this system is derived as the solution of a Maximum Weighted Bipartite Matching (MWBM) problem, where, roughly speaking, each of the N transmit antennas is assigned to a different user. The MWBM problem inspires a user selection algorithm where a subset of N out of K users is served. The proposed algorithm runs in polynomial-time (rather than involving an exhaustive search among all possible subsets of size N out of K users) and extends the classical DoF analysis to more realistic wireless channel configurations where users can experience very different channel gains from the base station. Extensive numerical simulations, run in practically relevant Rayleigh fading environments for different numbers of users and of antennas, show that the throughput achieved by serving the set of N users selected by the MWBM-based algorithm is at most N log(K) bits away from an outer bound to the sum-capacity, where in principle all the K users are served. Comparisons with another widely used user scheduling algorithm are also provided.