We consider the problem of assigning agents to slots on a line, where only one agent can be served at a slot and each agent prefers to be served as close as possible to his target. Our focus is on aggregate gap minimizing methods, i.e., those that minimize the total gap between targets and assigned slots. We first consider deterministic assignment of agents to slots, and provide a direct method for testing if a given deterministic assignment is aggregate gap minimizing. We then consider probabilistic assignment of agents to slots, and make use of the previous method to propose an aggregate gap minimizing modification of the classic random priority method to solve this class of problems. We also provide some logical relations in our setting among standard axioms in the literature on assignment problems, and explore the robustness of our results to several extensions of our setting.