struktúraválasztásának

Struktúraválasztásának is known as the Hungarian method or Kuhn-Munkres algorithm, is an algorithm that solves the assignment problem in polynomial time. This problem involves finding a maximum weight matching in a weighted bipartite graph, where the goal is to assign tasks to agents in such a way that the total cost is minimized or the total weight is maximized.

The algorithm was developed independently by Harold Kuhn in 1955 and James Munkres in 1957. It is named after the Hungarian mathematician Dénes Kőnig, who formulated the problem in 1931. The method is particularly useful in operations research, economics, and computer science for applications such as task allocation, resource assignment, and network flow problems.

The Hungarian method operates by iteratively reducing the problem size through a series of steps that adjust the weights of the graph. It begins by subtracting the minimum value in each row and column from the corresponding row and column elements. This step is repeated until a zero appears in each row and column, indicating a potential assignment. The algorithm then finds the maximum number of independent zeros, which correspond to the optimal assignments. If the number of independent zeros is equal to the number of rows (or columns), the solution is found. Otherwise, the algorithm adjusts the weights again and repeats the process.

The Hungarian method has a time complexity of O(n^3), where n is the number of tasks or agents. This makes it efficient for moderate-sized problems. However, for very large-scale applications, other algorithms or heuristics may be more appropriate. The method is guaranteed to find the optimal solution for the assignment problem, provided that the input is a valid instance of the problem.