Solving the Many to Many assignment problem by improving the Kuhn–Munkres algorithm with backtracking

• It improves the K–M algorithm to solve the M–M assignment problem.
• The improved algorithm (KMB) introduces backtracking.
• The KMB algorithm is valid and the worst time complexity is O((∑La[i])3)O((∑La[i])3).
• It provides the necessary and sufficient conditions for the solution.
• It illustrates the validity and efficiency of the KMB algorithm through simulations.

The Many to Many (M–M) assignment problem is an important open problem where one task is assigned to many, but different, agents and one agent may undertake many, but different, tasks. The Kuhn–Munkres (K–M) algorithm is a famous and traditional process used in dealing with assignment problems. In this paper, we propose a solution to the M–M assignment problem by improving the K–M algorithm with backtracking (KMB). To demonstrate the solution's suitability, we prove that the proposed KMB algorithm is valid and that the worst time complexity of the KMB algorithm is O((∑La[i])3)O((∑La[i])3), where La[i]La[i] denotes the maximum number of tasks that can be assigned to agent i. After that, we discuss several critical problems related to the algorithm and provide the necessary and sufficient conditions of solving the M–M assignment problem. Finally, we demonstrate, through experimentation, the validity, practicality and efficiency of the KMB algorithm.