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Slides9

# Slides9 - Lecture 9 Maximum Flow Problems(continued...

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1 Lecture 9 Maximum Flow Problems (continued) Ford-Fulkerson method Define I = {(i,j) E:±x ij < c ij } – arcs currently not “saturated” R = {(i,j) E:±x ij > 0} – arcs currently not “empty” We can increase the flow along arcs in I, and we can reduce the flow along arcs in R while still satisfying the capacity constraints 0 x ij c ij . Start with the feasible flow: x ij =0 for all (i,j) E (this is a feasible flow with value 0). The Ford-Fulkerson Method gives a way to maintain also the flow conservation.

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2 Ford-Fulkerson method Step 0: Set all nodes to be unlabeled and unscanned. Label node s with (*, ). Step 1: Choose any labeled but unscanned node, say i V with label (p,l(i)). For (i,j) I and j unlabeled: Set l(j)=min{l(i),c ij -x ij } and label j with (i + ,l(j)) For (j,i) R and j unlabeled: Set l(j)=min{l(i),x ji } and label j with (i - ,l(j)) SCAN Repeat Step 1 until t is labeled (then go to Step 2), or no more labeling is possible – STOP, flow is maximal. Step 2: Use labels to adjust flow, working backwards from t to s. If t is labeled (i
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Slides9 - Lecture 9 Maximum Flow Problems(continued...

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