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Our goal is to complete all jobs minimizing total inputs, while assigning each worker to exactly one job and vice versa. Small example just to make things clearer: This problem is known as the assignment problem.The assignment problem is a special case of the transportation problem, which in turn is a special case of the min-cost flow problem, so it can be solved using algorithms that solve the more general cases.The main idea of the method is the following: consider we’ve found the perfect matching using only edges of weight 0 (hereinafter called “0-weight edges”).
In order to avoid this, on each step we can just modify the matching from the previous step, which only takes O(n2) operations.
It’s easy to see that no more than n2 iterations will occur, because every time at least one edge becomes 0-weight. This is simply a function (for each vertex we assign some number called a label).
If we can’t find perfect matching on the current step, then the Hungarian algorithm changes weights of the available edges in such a way that the new 0-weight edges appear and these changes do not influence the optimal solution.
To clarify, let’s look at the step-by-step overview: Step 0)A. Apply the same procedure for the vertices in the right part (jobs). Otherwise find the minimum vertex cover (for the subgraph with 0-weight edges only), the best way to do this is to use Köning’s graph theorem.
be a spanning subgraph of G (in other words, it includes all vertices from G).
If G only those edges (x,y) which satisfy the following condition: , then it is an equality subgraph.There are also implementations of Hungarian algorithm that do not use graph theory.Rather, they just operate with cost matrix, making different transformation of it (see  for clear explanation).In other words, it only includes those edges from the bipartite matching which allow the vertices to be perfectly feasible.Now we’re ready for the theorem which provides the connection between equality subgraphs and maximum-weighted matching: is called alternating if its edges alternate between M and E\M.In this article we’ll deal with one optimization problem, which can be informally defined as: Assume that we have N workers and N jobs that should be done. Let’s look at the job and workers as if they were a bipartite graph, where each edge between the .For each pair (worker, job) we know salary that should be paid to worker for him to perform the job. Then our task is to find minimum-weight matching in the graph (the matching will consists of N edges, because our bipartite graph is complete).See picture below for explanation: At last, here's the function that implements Hungarian algorithm: To see all this in practice let's complete the example started on step 0.| Build alternating tree V| Augmenting V path found| Build V alternating tree| Update V labels(Δ=1)| Build alternating V tree| Update V labels(Δ=1)→| Build V alternating tree| Augmenting V path found| Build V alternating tree| Update labels V (Δ=2)| Build V alternating tree| Update labels V (Δ=1)| Build V alternating tree| Augmenting V path found Optimalmatching found Finally, let's talk about the complexity of this algorithm.On each iteration we increment matching so we have n iterations.Let’s call this labeling feasible if it satisfies the following condition: .In other words, the sum of the labels of the vertices on both sides of a given edge are greater than or equal to the weight of that edge.