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This study proposes two new dynamic assignment algorithms to match refugees and asylum seekers to geographic localities within a host country. The first, currently implemented in a multi-year randomized control trial in Switzerland, seeks to maximize the average predicted employment level (or any measured outcome of interest) of refugees through a minimum-discord online assignment algorithm. The performance of this algorithm is tested on real refugee resettlement data from both the US and Switzerland, where we find that it is able to achieve near-optimal expected employment compared to the hindsight-optimal solution, and is able to improve upon the status quo procedure by 40-50%. However, pure outcome maximization can result in a periodically imbalanced allocation to the localities over time, leading to implementation difficulties and an undesirable workflow for resettlement resources and agents. To address these problems, the second algorithm balances the goal of improving refugee outcomes with the desire for an even allocation over time. We find that this algorithm can achieve near-perfect balance over time with only a small loss in expected employment compared to the employment-maximizing algorithm. In addition, the allocation balancing algorithm offers a number of ancillary benefits compared to pure outcome maximization, including robustness to unknown arrival flows and greater exploration.