{
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  "body": "\n## Statement\n\nLet $D = (V, E)$ be a finite [[directed graph|0R5HTV7REV]] with a source $s$, a sink $t$ and a capacity function $c$ on the edges. Let $\\phi$ be a flow of value $v$ on this network. A flow obeys $0 \\le \\phi(e) \\le c(e)$ on each edge, and it is conserved at each vertex other than $s$ and $t$.\n\nLet $P = (x_0, x_1, \\dots, x_m)$ be an augmenting path from $s$ to $t$. Each step of $P$ uses an edge of $D$ either forwards or backwards. A forward edge has spare capacity $c(e) - \\phi(e) \u003e 0$. A backward edge carries positive flow $\\phi(e) \u003e 0$. Let $\\delta \u003e 0$ be the smallest of the spare capacities of the forward edges and the flows of the backward edges on $P$.\n\nDefine a new function $\\hat\\phi$ as follows. Add $\\delta$ to $\\phi$ on each forward edge of $P$. Subtract $\\delta$ from $\\phi$ on each backward edge of $P$. Leave $\\phi$ unchanged on all other edges.\n\nThen $\\hat\\phi$ is a flow on the network, and its value is $v + \\delta$.\n\n## Remarks\n\nThe choice of $\\delta$ keeps every edge value between $0$ and its capacity. At each inner vertex of $P$ the two changes cancel, so conservation still holds. At $s$ the net outflow grows by exactly $\\delta$.\n",
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    "name": "Augmenting along a path raises the flow value by delta",
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        "locator": "source/s_networkflow_augmenting.ptx:45",
        "url": "https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_networkflow_augmenting.ptx#L45"
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  "name": "Augmenting along a path raises the flow value by delta",
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