Upper and lower bounds network flow and cost flow study notes

Foreword

Good article:
https://www.cnblogs.com/liu-runda/p/6262832.html
https://www.cnblogs.com/leason-lyx/p/11144527.html

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Passive exchange bounds feasible flow:

Model: a network, a stream is obtained, so that each edge must flow $\ge L$ and $\ And R$ , each point must meet the total amount of inflow outflow = (flow conservation) (this feature is that the flow cycle, without beginning or end)

Taking into account if there is a feasible flow, the flow rate of each side is at least $L$ , so we can advance it first stream $L$

Such inflows are likely after a point is not the amount of outflow

We consider obtaining an additional flow, and ultimately flow to each side plus an additional stream flow $L$

Specifically, we completed the construction side $(u,v,R-L)$

Finally, let consider the flow balance, classified Discussion:
If the initial inflow outflow = a balance point it
if the initial inflow> outflow, then the additional flow in the inflow point <outflow, called $A$ point-like
if the initial inflow <outflow, then the additional flow into> outflow, called $B$ class point
referred to as the difference $|\Delta|$

In order to balance the flow, we have established a super sources and sinks $S,T$， $S$ to $B$ class side point even flow rate of $|\Delta|$
$A$ point-like to $T$ even edge, flow $|\Delta|$

Easy to find, source and Meeting Point $S,T$ to connect Collage rights and are equal
solvable if and only if the imbalance of traffic filled, that is, $S,T$ all the side edges of the full flow

To run a $S$ to $T$ maximum flow to the

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The active exchange bounds feasible flow

Traffic source and the sink does not always balance

Considering the source of the flow out into the flow = sink, the sink even flow source $+\infty$ edges can be balanced

The entire traffic flow is feasible flow source to sink that edge

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There is an active exchange bounds maximum flow

The maximum flow rate that is feasible under conditions of flow balance

Construction is shown above, we found that some residual network is not run over the edge

The source and sink side with super disconnected, then ran a picture on the residual network $S,T$ Maximum flow

The final maximum flow need to add the original feasible flow

The minimum flow can be reversed run, $T$ to $S$

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Passive exchange bounds minimum cost feasible flow

Also build super sources and sinks $S,T$

For edge $(u,v,L,R,cost)$ , built edge $(u,v,R-L,cost)$ , and the initial value plus $L*cost$

Point for traffic imbalances built side "feasible flow of passive sinks lower and upper bounds," with a cost of 0

Then run $S$ to $T$ minimum cost maximum flow

Add a picture in the active exchange $(T,S,\infty,0)$