erved. Any use is subject to the Terms of Use as given at the website. Capacity and Quality Tradeoffs Capacity and Quality Principles Figure 22.2 Flow network. Mathematically speaking, this is a combinatorial graph defined and described in Sec. 26.1.2. Each dot is a node or vertex, and each arrow is an arc or edge. The arcs have maximum flow capacities, and the object of the network flow game is to get as much stuff to flow from source S to target T as possible. The network flow structure is par- ticularly amenable to computational solution, so mathematical modelers work hard to put practical problems into the network flow model. As a result, network flow compu- tations can get quite large, some with millions of arcs. We start solving the network flow problem by finding routes from source S to target T that have flow capacity available. We add flow along those routes until they can han- dle no more, until some of their arcs are saturated. We think of these as bottlenecks in the solution process. Rather than give up at the first bottleneck, we look for alternate routes from source S to target T, other ways to increase flow. Sometimes we find that we have to decrease flow along one route so that we can use capacity more effectively on another route. At some point in the solution process, it becomes impossible to add more flow from source S to target T. There is a set of arcs that are at maximum capacity, final bottle- necks that are called critical resources in the solution. The other resources are called slack resources. Expanding the critical resources can improve the flow in the network, whereas expanding the slack resources has no effect on the solution. It is the job of net- work designers to find ways to add capacity to the critical resources. Sometimes the in- ternal flows have their own costs so that the solution is a tradeoff between the value of the flow and the cost of the resources. The power of the network flow problem is that the flows can be metaphors for a wide variety of things. Some problems invol