Makespan-minimized multi-agent path planning (MAPP) seeks to minimize the time taken by the slowest of n agents to reach its destination and this is essentially a minimax-constrained optimization problem. In this work, an iterative max-min improvement (IMMI) algorithm is proposed to approximate the optimal solution of the makespan-minimized MAPP problem. At each iteration, a linear maximization problem is solved using a simplex method followed by a computationally hard MAPP minimization problem that is solved using a local search approach. To keep the local search from being trapped in an unfeasible solution, a Guided Local Search technique is proposed. Comparative results with other MAPP algorithms suggest that the proposed IMMI algorithm strikes a good tradeoff between the ability to find feasible solutions that can be traversed quickly and the computational time incurred in determining these paths.