Capacity
Named by 4 essays across one field — each of them below, with the objects they name alongside it.
The bottleneck is the whole story
However much a network can carry from one place to another, there is a way of cutting it in two whose total capacity is exactly that number. One quantity is a maximum over ways of routing and the other a minimum over ways of severing, and they are never off by even one.
The cheapest way to send
Put a price on every road as well as a capacity and ask for the cheapest way to send four units. Twenty-eight ways exist and one is cheapest, and two certificates prove it without comparing it with the other twenty-seven: no cycle of roads it leaves unused costs less than nothing to push round, and there are prices at the places that every usable road fails to beat.
A search for room that never ends
The textbook way to find the most that a network can carry is to find any route that still has room, push as much along it as it will take, and repeat until no route has room left. With whole-number capacities that must stop. Give one road a capacity of 0.618…, the golden ratio's reciprocal, and choose the routes badly, and it never stops: every push is positive, the flow converges, and it converges to 4.236 when the network can carry 5. The same choice made carelessly in floating point does stop — because of rounding, at the wrong answer, convinced it has finished.
Out before the table says so
Detroit has 49 wins and 27 games left, so it can reach 76, and the leader has 75. On paper it is alive. It is not: the four teams ahead must play 27 games among themselves, and whoever wins them, those four finish with an average above 76. Finding that takes one maximum flow, and it is checked against every possible result of every game.
Named alongside it
The objects these essays reach for when they reach for this one.
FlowCertificateCutMin-max theoremAlgorithmBipartite matchingConservationConvexityCycleDualityEdge-disjoint pathsExhaustive search