Homelessness is a humanitarian challenge affecting an estimated 1.6 billion people worldwide. In the face of rising homeless populations in developed nations and a strain on social services, government agencies are increasingly adopting data-driven models to determine one's risk of experiencing homelessness and assigning scarce resources to those in need. We conducted a systematic literature review of 57 papers to understand the evolution of these decision-making algorithms. We investigated trends in computational methods, predictor variables, and target outcomes used to develop the models using a human-centered lens and found that only 9 papers (15.7%) investigated model fairness and bias. We uncovered tensions between explainability and ecological validity wherein predictive risk models (53.4%) focused on reductive explainability while resource allocation models (25.9%) were dependent on unrealistic assumptions and simulated data that are not useful in practice. Further, we discuss research challenges and opportunities for developing human-centered algorithms in this area.