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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 [1] for clear explanation).be a spanning subgraph of G (in other words, it includes all vertices from G).

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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.

We’ll not touch these approaches, because it’s less practical for Top Coder needs.

As mentioned above, we are dealing with a bipartite graph.

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).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).For each vertex from left part (workers) find the minimal outgoing edge and subtract its weight from all weights connected with this vertex. Actually, this step is not necessary, but it decreases the number of main cycle iterations. Find the maximum matching using only 0-weight edges (for this purpose you can use max-flow algorithm, augmenting path algorithm, etc.). Step 2) Let and adjust the weights using the following rule: Step 3) Repeat Step 1 until solved.But there is a nuance here; finding the maximum matching in step 1 on each iteration will cause the algorithm to become O(n5). Wiley Online Library requires cookies for authentication and use of other site features; therefore, cookies must be enabled to browse the site.Detailed information on how Wiley uses cookies can be found in our Privacy Policy.We’ll handle the assignment problem with the Hungarian algorithm (or Kuhn-Munkres algorithm).I’ll illustrate two different implementations of this algorithm, both graph theoretic, one easy and fast to implement with O(n4) complexity, and the other one with O(n3) complexity, but harder to implement.Discuss this article in the forums Are you familiar with the following situation?You open the Div I Medium and don’t know how to approach it, while a lot of people in your room submitted it in less than 10 minutes.