Graph_Theory_Notes5.pdf

# Max flow min cut theorem this is a fundamental result

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Max-Flow Min-Cut Theorem This is a fundamental result in graph theory and combinatorial opti- mization. Theorem 1 (Ford and Fulkerson 1956) . In every network, the value of a maximum flow equals the capacity of a minimum cut. Proof. Let f be a maximum flow in a network N . Then there is no f - augmenting path, for otherwise there is a flow with larger value. Let S := { w V : there is an f -unsaturated path from x to w } . Then x S and y 6∈ S , and so ( S, V - S ) is a cut. For each uv ( S, V - S ), we have f ( uv ) = c ( uv ), for otherwise adding uv to an f -unsaturated ( x, u )- path yields an f -unsaturated ( x, v )-path, which contradicts the assumption that v V - S . Similarly, for each arc uv from V - S to S , we have f ( uv ) = 0. By previous lemmas, val ( f ) = f + ( S ) - f - ( S ) = cap ( S, V - S ) . Hence f is a maximum flow and ( S, V - S ) is a minimum cut.

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Theorem 2. A flow f is maximum iff there exists no f -augmenting path. Proof. If f is maximum, then there is no f -augmenting path, for otherwise we can augment along this path to get a new flow with larger value. Con- versely, if there is no f -augmenting path, then the proof of the Max-Flow Min-Cut Theorem gives a cut ( S, V - S ) such that val ( f ) = cap ( S, V - S ). Hence f is a maximum flow. This theorem is the basis for the following maximum flow algorithm. 3 Maximum flow algorithm Labelling algorithm The label on each vertex v denotes I ( P ) for some path P from x to v . 1. Let f ( a ) = 0 for every arc a . 2. Let the label of x be ( x ) := . repeat (i) Choose an unsaturated arc a = uv whose tail u is labelled and whose head v is unlabelled. Label v by ( v ) := min { ( u ) , c ( a ) - f ( a ) } . OR (ii) Choose an arc a = vu which is not f -zero, with head u labelled and tail v unlabelled. Label v by ( v ) := min { ( u ) , f ( a ) } . (In (i) and (ii) we say v was labelled from u .) until no arc a as in (i) or (ii) can be found or y gets labelled. 3. If y is unlabelled, go to step 4. Otherwise, update the flow f by augmenting extra flow ( y ) along a path P from x to y , where P can be found by defining a tree T containing all arcs uv or vu such that v was labelled from u ; then P is the ( x, y )-path in T . Define a new flow ˆ f by ˆ f ( a ) = f ( a ) + ( y ) if a is forward in P f ( a ) - ( y ) if a is backward in P f ( a ) if a is not in P. Reset all the labels, set f := ˆ f and return to STEP 2. 4. Output the maximum flow f . The set S of labelled vertices determines a minimum cut ( S, V - S ). Proof of the labelling algorithm Let S be the set of labelled vertices at the end of the algorithm . Then x S since x is labelled by , and y 6∈ S for otherwise the algorithm would not stop. Hence ( S, V - S ) is a cut. We claim that every arc in ( S, V - S ) (i.e. from S to V - S ) is saturated by the final flow f . Suppose otherwise. Let uv ( S, V - S ) be unsaturated. Since u S , u is labelled. Since uv is unsaturated, by rule 2(i) we should label v from u , which contradicts the assumption v V - S .
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