*But then M contains two edges that share the vertex u, contradicting the fact that M is a matching.*Thus every edge in M has either both endpoints or neither endpoint in Z.

*But then M contains two edges that share the vertex u, contradicting the fact that M is a matching.*Thus every edge in M has either both endpoints or neither endpoint in Z.

Step 1: Subtract row minima We start with subtracting the row minimum from each row.In each step, either we modify y so that its value increases, or modify the orientation to obtain a matching with more edges.We maintain the invariant that all the edges of M are tight. In a general step, let increases (note that the number of tight edges does not necessarily increase).We consider an example where four jobs (J1, J2, J3, and J4) need to be executed by four workers (W1, W2, W3, and W4), one job per worker.The matrix below shows the cost of assigning a certain worker to a certain job.For example: The Hungarian method, when applied to the above table, would give the minimum cost: this is , achieved by having Paul clean the bathroom, Dave sweep the floors, and Chris wash the windows.There are two ways to formulate the problem: as a matrix or as a bipartite graph.Fill in the cost matrix of an assignment problem and click on 'Solve'.The optimal assignment will be determined and a step by step explanation of the hungarian algorithm will be given.The Hungarian method finds a perfect matching and a potential such that the matching cost equals the potential value. In fact, the Hungarian method finds a perfect matching of tight edges: an edge ) which has the property that the edges oriented from T to S form a matching M.Initially, y is 0 everywhere, and all edges are oriented from S to T (so M is empty).

## Comments Hungarian Method Solving Assignment Problem

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