MATLAB Implementation of the Hungarian Algorithm for Optimal Assignment Solution

The Hungarian Algorithm is a classic combinatorial optimization algorithm primarily used to solve assignment problems. It finds the optimal person-task assignment in a given cost matrix to minimize total cost. Developed by Hungarian mathematician Kuhn, this algorithm is based on the maximum weight matching principle in graph theory. Implementing the Hungarian Algorithm in MATLAB typically involves the following key computational steps: First, construct a two-dimensional matrix representing costs between tasks and personnel. The algorithm uses row and column transformations to find independent zero elements through two main operations: row reduction and column reduction. During row reduction, each row's elements are subtracted by the row's minimum value; column reduction performs similar operations on each column. The implementation then uses the line-covering method (drawing lines) to verify whether the optimal solution has been found - covering all zero elements with the minimum number of horizontal or vertical lines. If the number of lines equals the matrix dimension, the optimal solution is obtained; otherwise, further matrix adjustments are required until optimal conditions are met. For MATLAB implementation, the algorithm requires handling matrix operations and loop judgments while efficiently finding combinations of independent zero elements. A robust implementation should process matrices of different dimensions and properly handle exceptional cases (such as no-solution scenarios). Key MATLAB functions involved would include min() for reduction operations, matrix indexing for zero element identification, and while loops for iterative line-covering checks. The implementation typically uses logical arrays to track covered rows/columns and requires careful management of temporary matrices during adjustment phases.