====== Chapter 7 =====

===== 7.1 The maximum - flow problem and the ford - fulkerson Algorithm =====
  * want to model a flow network 
  * associated with edges are capacity, which is non negative
  * single source node s 
  * single sink node t
  * Need to define flow
  * need to figure out how to get maximum flow over network
  * Need a residual graph
  * algorithm on p. 342
  * algorithm O(mC) 
===== 7.2 Maximum flows and minimum cuts in a network =====
  * let f be any s-t flow, and (A, B) any s-T cut then V(f) = Fout(a) - fin(a) 
  * proofs of why this is maximum flow on 347
  * max flow = min cut
  * 7.9 - if f is an s-t flow sucht that there is no s-t path in the residual graph Gf, then there is an s-t cut (a*, b*) in G for which v(F) = c(A*,B*) consequently f has maximum value of any flow in G, and (A*, B*) has the minimum capacity of any s-t cut in G. 
  * 7.10 - the flow f returned by the ford - fulkerson algorithm is a maximum flow. 
  * 7.11 given a flow f of maximum value, we can compute as s-t cut of minimum capacity in O(m) time. 
  * 7.13 - in every flow network, the maximum value of an s-t flow is equal to the minimum capacity of an s-t cut. 7.14 if all capacities in the flow network are integers, then there is a maximum flow f for which every flow value f(e) is an integer. 

===== 7.5 A first application: The bipartite matching problem =====
  * going back to our bipartite problem, but there is a source node and a sink node. and the bipartite two teams are split one with the source node and one with the sink node
  * 7.34 m' contains k edges
  * each node in x is the tail of at most one edge in m'
  * each node in y is the head of at most one edge in m'
  * the size of the maximum matching in G is equal to the value of the maximum flow in G' and hte edges in such a matching in G are the edges that carry flow from x to y in G. 
  * can use the ford fulkerson to find this maximum matching in bipartite graphs in O(mn) time
===== 7.7 Extensions to the maxmim flow problem =====
The problem
  * Circulations with demands 
  * we have demand nodes and supply nodes 
  * the supply has negative deamnd 

design the algorithm
  * put all the supply nodes with one source node 
  * put all the demand nodes with the one sink node
  * pretty simple if we think of it that way. 

===== 7.8 survey design =====
The problem
  * questions for consumers abotu products
  * only a certain number of questions for each consumer
  * only a certain number of customers have each product 
  * more details on the algorithm on page 286
  * images are so helpful on 386



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