Abstract In recent years, a number of new challenges have been observed in the application of matching theory. One of the most pressing problems concerns how to allocate refugees to hosts safely and in a timely manner. Currently, this placement is implemented on an ad hoc basis where the preferences of both refugees and hosts are not taken into account. Another important realization is that reallife matching markets are often subject to various distributional constraints. For example, there has been increased attention to school choice models that take account of affirmative action and diversity concerns. The objective of this research is to design efficient algorithms while satisfying desirable properties for these new emerging problems