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Unformatted text preview: Algorithms Lecture 23: Applications of Maximum Flow [ Fa’10 ] For a long time it puzzled me how something so expensive, so leading edge, could be so useless, and then it occurred to me that a computer is a stupid machine with the ability to do incredibly smart things, while computer pro grammers are smart people with the ability to do incredibly stupid things. They are, in short, a perfect match. — Bill Bryson, Notes from a Big Country (1999) 23 Applications of Maximum Flow 23.1 EdgeDisjoint Paths One of the easiest applications of maximum flows is computing the maximum number of edgedisjoint paths between two specified vertices s and t in a directed graph G using maximum flows. A set of paths in G is edgedisjoint if each edge in G appears in at most one of the paths; several edgedisjoint paths may pass through the same vertex, however. If we give each edge capacity 1, then the maxflow from s to t assigns a flow of either 0 or 1 to every edge. Since any vertex of G lies on at most two saturated edges (one in and one out, or none at all), the subgraph S of saturated edges is the union of several edgedisjoint paths and cycles. Moreover, the number of paths is exactly equal to the value of the flow. Extracting the actual paths from S is easy—just follow any directed path in S from s to t , remove that path from S , and recurse. Conversely, we can transform any collection of k edgedisjoint paths into a flow by pushing one unit of flow along each path from s to t ; the value of the resulting flow is exactly k . It follows that the maxflow algorithm actually computes the largest possible set of edgedisjoint paths. The overall running time is O ( V E ) , just like for maximum bipartite matchings. The same algorithm can also be used to find edgedisjoint paths in undirected graphs. We simply replace every undirected edge in G with a pair of directed edges, each with unit capacity, and compute a maximum flow from s to t in the resulting directed graph G using the FordFulkerson algorithm. For any edge uv in G , if our max flow saturates both directed edges u v and v u in G , we can remove both edges from the flow without changing its value. Thus, without loss of generality, the maximum flow assigns a direction to every saturated edge, and we can extract the edgedisjoint paths by searching the graph of directed saturated edges. 23.2 Vertex Capacities and VertexDisjoint Paths Suppose we have capacities on the vertices as well as the edges. Here, in addition to our other constraints, we require that for any vertex v other than s and t , the total flow into v (and therefore the total flow out of v ) is at most some nonnegative value c ( v ) . How can we compute a maximum flow with these new constraints?...
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This note was uploaded on 10/14/2011 for the course ECON 101 taught by Professor Smith during the Spring '11 term at West Virginia University Institute of Technology.
 Spring '11
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