What is the Hungarian algorithm for solving the bipartite matching problem in a graph?

The Hungarian algorithm is a combinatorial optimization algorithm used to solve the bipartite matching problem in a graph. It was developed by Hungarian mathematicians in the 1930s and is also known as the Kuhn-Munkres algorithm.

The bipartite matching problem involves finding a maximum cardinality matching in a bipartite graph, where the graph is divided into two disjoint sets of vertices, often referred to as the "left" and "right" sets. The goal is to find a matching that pairs as many vertices as possible, with each vertex being paired with at most one vertex from the other set.

The Hungarian algorithm follows a step-by-step process to find the maximum cardinality matching. Here is a brief overview of the algorithm:

1. Initialize a matrix called the cost matrix, which represents the costs associated with matching each vertex from the left set to a vertex from the right set. If a vertex cannot be matched, the cost is set to infinity.

2. Apply the following steps until a maximum matching is found:
a. Find the minimum cost element in each row of the cost matrix and subtract it from all other elements in the same row.
b. Find the minimum cost element in each column of the modified cost matrix and subtract it from all other elements in the same column.
c. Cover the matrix with the minimum number of lines (horizontal or vertical) such that all zeros in the matrix are covered. If the number of lines is equal to the number of vertices, a maximum matching is found. Otherwise, proceed to the next step.
d. Find the smallest uncovered element in the matrix and subtract it from all uncovered elements. Add it to all elements covered by two lines. Go back to step c.

3. Once a maximum matching is found, the pairs of matched vertices can be determined by examining the matrix. Each row with a non-zero element represents a matched vertex pair.

The Hungarian algorithm guarantees finding the maximum cardinality matching in a bipartite graph and has a time complexity of O(n^3), where n is the number of vertices in the graph. It is widely used in various applications, such as assignment problems, scheduling, and resource allocation.