Maximum Flow

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<ul><li><p>Maximum Flow</p></li><li><p>The Standard Maximum Flow Problem</p><p> (flow-capacity) , </p><p> X Y . , </p></li><li><p>Rephrasing in terms of graph theory</p><p> - (directed graph)</p><p> (capacity) c X, Y , </p><p> (flow) f </p><p> f </p></li><li><p>Residual networks</p><p> . : f c </p><p> : f </p></li><li><p>Augmenting paths</p><p> Residual network </p><p> (path capacity) </p></li><li><p>How to Solve It (1/N)</p><p> Figure 2b Residual network Augmenting path? X-A-C-Y</p><p> C-Y 1</p><p> 1 </p></li><li><p>How to Solve It (2/N)</p><p> Figure 3b Residual network Augmenting path? X-A-C-B-D-E-Y</p><p> C-B 1</p><p> 1 </p></li><li><p>How to Solve It (3/N)</p><p> Figure 1b Residual network Augmenting path? </p></li><li><p>The Ford-Fulkerson method</p><p>//Assumption 1: capacities and flows of the edges being integers </p><p>//Assumption 2: path-capacity being positive </p><p>Begin </p><p>x := 0</p><p>create the residual network G(x)</p><p>while there is some augmenting path from s to t in G(x) </p><p>begin</p><p>let P be a path from s to t in G(x)</p><p>min:= Capacity of P</p><p>Increase flow by min along P</p><p>Update G(x)</p><p>end</p><p>End { the flow x is now maximum }</p></li><li><p>Correctness of the Ford-Fulkerson </p><p>method Augmenting path maximum flow?</p><p> Maximum flow Augmenting path </p><p> Maximum flow </p><p> Augmenting path maximum flow. </p></li><li><p>Choosing Good Augmenting Paths</p><p> Ford-Fulkerson algorithm augmenting path maximum flow </p><p> augmenting path 1 </p><p> (shortest path)</p><p> (maximum path capacity)</p></li><li><p>Reduction of multiple-source/ </p><p>multiple sink </p><p> dummy vertex</p><p> super-source: . </p><p> super-sink: . .</p></li><li><p>Eliminating vertex-capacities</p><p> 2, </p><p> , </p></li><li><p>Maximum Bipartite Matching</p><p> 11 ( X, Y )</p><p> super-source: 1. 1 </p><p>A B </p><p> super-sink</p></li></ul>