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MCMF cleaner proof of cost when applying Johnson potentials #376

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Update mincost-maxflow.md
Try2Take Jan 23, 2026
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Update mincost-maxflow.md
Try2Take Jan 23, 2026
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Update mincost-maxflow.md
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Update mincost-maxflow.md 
fix
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Try2Take authored Jan 23, 2026
commit 2770a09144e02cf722f4f403461d2b1a2804267b
2 changes: 1 addition & 1 deletion content/russian/cs/flows/mincost-maxflow.md
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Expand Up @@ -53,7 +53,7 @@ $$ w_{uv}' = w_{uv} + d_u - d_v $$
**Доказательство.** Пусть вес какого-то ребра $(u, v)$ отрицателен, то есть $w_{uv}' = w_{uv} + d_u - d_v < 0$. Тогда $d_u + w_{uv} < d_v$, и нарушилось неравенство треугольника: почему мы тогда не использовали ребро $(u, v)$, когда искали кратчайший путь до $v$?

Аналогично можно показать, что рёбра на кратчайших путях из $s$ имеют нулевую стоимость. Заметим, что стоимость *обратных* рёбер на кратчайших путях тоже будет нулевой. Возьмём ребро $(u, v)$, которое лежит на кратчайшем пути из $s$, то есть $d_v = d_u + w_{uv}$. Тогда:
$$ w_{vu}' = -w_{uv} + d_v - d_u = -w_{uv} + (d_u + w_{uv}) - d_u = -w_{uv} + w_{uv} = 0 $$
$$w_{vu}' = -w_{uv} + d_v - d_u = -w_{uv} + (d_u + w_{uv}) - d_u = -w_{uv} + w_{uv} = 0$$

**Утверждение 2**. Кратчайшие пути между любыми вершинами остались кратчайшими.

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