The **Birkhoff polytope** *B*_{n}, also called the **assignment polytope**, the **polytope of doubly stochastic matrices**, or the **perfect matching polytope** of the complete bipartite graph, is the convex polytope in **R***N* (where *N* = *n*²) whose points are the doubly stochastic matrices, i.e., the *n* × *n* matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1.

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