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  "body": "\n## Summary\n\nA totally unimodular matrix is an integer matrix in which every square submatrix has determinant $0$, $1$ or $-1$.\nIn particular every entry is $0$, $1$ or $-1$.\nA linear program whose constraint matrix is totally unimodular and whose right-hand side is integral has integral vertices.\nThis explains why many problems on graphs, such as bipartite matching and network flows, can be solved exactly by linear programming.\n\n## Explanations\n\n### L1 human\n\nLinear programs are easy to solve, but problems that ask for whole-number answers are usually hard.\nFor some special constraint matrices, the two coincide: whenever the right-hand side is whole-number and the linear program has a best solution, it has a best solution in whole numbers.\nTotally unimodular matrices are exactly such special matrices: every square piece of them has determinant $0$, $1$ or $-1$.\nThe incidence matrix of a bipartite graph, with a row for each vertex and a column for each edge, is a standard example.\n",
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