Title: On Threshold Functions in Learning to Generate Feasible Solutions of Mixed Integer Programs. Abstract: Finding a high-quality feasible solution to a combinatorial optimization problem in a given time budget is a challenging task due to its discrete nature. Neural diving is a learning-based approach to generating partial assignments for the discrete variables in MIP. We find that there usually is a small range of selection rates which lead to feasible and optimal solutions; when too many parameters are selected, the solution space is too restricted to find a feasible solution; when too few parameters are selected, the solution space is too wide to efficiently find a feasible solution. Therefore, the choice of selection rate is the critical determinant of the Neural diving performance. In this context, we present theoretical insights that there exist threshold functions in feasibility and feasible optimality over the selection rate. Based on the theoretical foundations, we introduce a post-hoc method, and a learning-based approach to optimize the selection rate for partial discrete variable assignments in MIP more efficiently. A key idea is to jointly learn to restrict the selection rate search space, and to predict the selection rate in the learned search space that results in a high-quality feasible solution. MIP solver is integrated into the end-to-end learning framework. We suggest that learning a deep neural network to generate a threshold-aware selection rate is effective in finding high-quality feasible solutions more quickly. Experimental results demonstrate that our method achieves state-of-the-art performance in NeurIPS ML4CO datasets. In the workload apportionment dataset, our method achieves the optimality gap of 0.45%, which is around 10× better than SCIP, at the one-minute time limit.