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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).
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 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.
Then, after the contest, you find out in the editorial that this problem can be simply reduced to a classical one.
If yes, then this tutorial will surely be useful for you. We can also rephrase this problem in terms of graph theory.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).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.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.Also, our problem is a special case of binary integer linear programming problem (which is NP-hard).But, due to the specifics of the problem, there are more efficient algorithms to solve it.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.