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Copy file name to clipboardExpand all lines: src/graph/hungarian-algorithm.md
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-**The Problem of Potentials**. Given a matrix $A[1 \ldots n][1 \ldots m]$, it is required to find two arrays $u[1 \ldots n]$ and $v[1 \ldots m]$ such that, for any $i$ and $j$, $u[i] + v[j] \leq a[i][j]$ and the sum of elements of arrays $u$ and $v$ is maximum.<br>
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Knowing the Hungarian algorithm, the solution to this problem will not be difficult: the Hungarian algorithm just finds such a potential $u, v$ that satisfies the condition of the problem. On the other hand, without knowledge of the Hungarian algorithm, it seems almost impossible to solve such a problem.
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!!! info "Remark"
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!!! info "Remark"
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This task is also called the **dual problem** of the assignment problem: minimizing the total cost of the assignment is equivalent to maximizing the sum of the potentials.
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This task is also called the **dual problem** of the assignment problem: minimizing the total cost of the assignment is equivalent to maximizing the sum of the potentials.