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Max Flow Problem Given network N=(V,A), two nodes s,t of V, and capacities on the arcs: uij is the capacity on arc (i,j). Find non-negative flow fij for each arc (i,j) such that for each node, except s and t, the total incoming flow is equal to the total outgoing flow (flow conservation), such that fij is at most the capacity uij on arc (i,j); The total amount of flow going out of s (which is equal to the total amount of flow coming into t ) is maximized

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**the black numbers next to an arc is its capacity**

1 3 2 4 3 2 3 1 t 1 1 s 2 4 2 4 4 the black numbers next to an arc is its capacity

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**Set costs all other arcs at 0 The minimum cost flow circulation (Af=0) **

1 3 2 4 3 2 3 1 t 1 1 s 2 4 2 4 4 Cts= -1 Set costs all other arcs at 0 The minimum cost flow circulation (Af=0) maximises the s-t flow

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**the black numbers next to an arc is its capacity **

1 3 2 4 3 2 3 1 t 1 1 s 2 4 2 4 4 the black numbers next to an arc is its capacity Push flow over the path s-1-4-t The bottlenecks on this path are the edges {1,4} and {4,t}. So we can send flow 2 along this path

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**the black number next to an arc is its capacity **

1 3 2 4 2 3 2 2 3 1 t 1 1 s 2 2 4 2 4 4 the black number next to an arc is its capacity the green number next to an arc is the flow on it We can push another extra flow of 1 on the path s-1-3-t. The bottleneck is now {s,1} with remaining capacity 1.

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**the black number next to an arc is its capacity **

1 3 1 2 1 4 3 3 2 2 3 1 t 1 1 s 2 2 4 2 4 4 the black number next to an arc is its capacity the green number next to an arc is the flow on it Since {4,t} is on its capacity, the only path remaining through which we can send extra flow is s t. Here {4,3} is the bottleneck and thus we can send extra flow of 1

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**the black number next to an arc is its capacity **

1 3 1 2 2 4 3 3 2 2 3 1 t 1 1 1 s 2 2 1 4 2 1 4 4 the black number next to an arc is its capacity the green number next to an arc is the flow on it There are no s-t paths left on which we can send extra flow. So is this the maximal flow?

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**the residual graph blue arcs (i,j) are forward arcs (fij<uij)**

1 3 3 3 2 2 3 3 1 1 t 1 1 1 1 s 2 2 2 4 blue arcs (i,j) are forward arcs (fij<uij) green arcs (j,i) are backward arcs (fij>0) the blue number is the residual capacity of a blue arc the green number is the capacity of a green arc the black number is the original capacity of the arc

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**the residual graph red arcs form an s-t-path in the residual graph**

1 3 1 1 2 2 3 2 3 1 t 1 1 s 2 1 3 2 1 3 4 red arcs form an s-t-path in the residual graph and therefore a flow-augmenting path in the original network

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**the residual graph red arcs form an augmenting path**

1 3 1 1 2 2 3 2 3 1 t 1 1 s 2 1 3 2 1 3 4 red arcs form an augmenting path Augment the flow by the minimum capacity of a red arc, i.e, 1

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**the residual graph Augment the flow by the minimum capacity of a**

1 3 1 1 2 2 3 2 3 1 t 1 1 s 2 1 3 2 1 3 4 Augment the flow by the minimum capacity of a red arc, i.e, 1: - increase the flow by 1 on all arcs corresponding to forward red arcs - decrease the flow by 1 on all arcs corresponding to backward red arcs

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**the residual graph Augment the flow by the minimum capacity of a**

1 3 2 3 1 3 3 1 1 1 t 1 1 s 2 2 2 2 2 2 4 Augment the flow by the minimum capacity of a red arc, i.e, 1: - increase the flow by 1 on all arcs corresponding to forward red arcs - decrease the flow by 1 on all arcs corresponding to backward red arcs Construct the new residual graph

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**An s-t cut is defined by a set S of the nodes with **

1 3 2 4 2 2 3 1 t 1 1 s 2 4 2 4 4 An s-t cut is defined by a set S of the nodes with s in S and t not in S.

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**An s-t cut is defined by a set S of the nodes with **

1 3 2 4 2 2 3 1 t 1 1 s 2 4 2 4 4 S={s,1,2} An s-t cut is defined by a set S of the nodes with s in S and t not in S.

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**An s-t cut is defined by a set S of the nodes with **

1 3 2 4 2 2 3 1 t 1 1 s 2 4 2 4 4 S={s,1,2} An s-t cut is defined by a set S of the nodes with s in S and t not in S Size of cut S is the sum of the capacities on the arcs from S to N\S.

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**An s-t cut is defined by a set S of the nodes with **

1 3 2 4 2 2 3 1 t 1 1 s 2 4 2 4 4 C(S)=u13+u14+u24= 2+2+4=8 An s-t cut is defined by a set S of the nodes with s in S and t not in S Size of cut S is the sum of the capacities on the arcs from S to N\S.

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S1 1 3 2 4 2 2 3 1 t 1 1 s 2 4 2 4 4 C(S1)=us1+u24= 2+4=6

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**S2 S1 C(S1)=us1+u24= 2+4=6 C(S2)=u13+u43+u4t= 2+1+2=5 t s 1 3 2 4 2 2**

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**S2 C(S2)=u13+u43+u4t= 2+1+2=5 Max Flow ≤ Min Cut t s 1 3 2 4 2 2 3 1 1**

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**S2 C(S2)=u13+u43+u4t= 2+1+2=5 Max Flow ≤ Min Cut fs1+fs2 ≤ Min Cut ≤ 5**

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**= the residual graph S2 fs1+fs2=2+3=5 C(S2)=u13+u43+u4t= 2+1+2=5**

3 1 3 3 1 1 1 t 1 1 s 2 2 2 2 2 2 4 = fs1+fs2=2+3=5 C(S2)=u13+u43+u4t= 2+1+2=5 Max Flow = Min Cut

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**= the residual graph S2 fs1+fs2=2+3=5 C(S2)=u13+u43+u4t= 2+1+2=5**

3 1 3 3 1 1 1 t 1 1 s 2 2 2 2 2 2 4 = fs1+fs2=2+3=5 C(S2)=u13+u43+u4t= 2+1+2=5 Max Flow = Min Cut Theorem: Max Flow = Min Cut

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**= Proof sketch of Max Flow=Min Cut S2 fs1+fs2=2+3=5**

3 1 3 3 1 1 1 t 1 1 s 2 2 2 2 2 2 4 = fs1+fs2=2+3=5 C(S2)=u13+u43+u4t= 2+1+2=5 There is no s-t-path in the residual graph!!! S2={s,1,2,4} the set of nodes reachable from S

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**= Proof sketch of Max Flow=Min Cut S2 fs1+fs2=2+3=5**

2 2 4 3 2 2 3 0 2 1 1 0 t 1 1 1 0 s 2 2 4 3 2 4 2 4 = fs1+fs2=2+3=5 C(S2)=u13+u43+u4t= 2+1+2=5 from S2={s,1,2,4} to {3,t} f13 = u13 = 2 f43 = u43 = 1 f4t = u4t = 2

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**= Proof sketch of Max Flow=Min Cut S2 fs1+fs2=2+3=5**

2 2 4 3 2 2 3 0 2 1 1 0 t 1 1 1 0 s 2 2 4 3 2 4 2 4 = fs1+fs2=2+3=5 C(S2)=u13+u43+u4t= 2+1+2=5 from S2={s,1,2,4} to {3,t} f13 = u13 = 2 f43 = u43 = 1 f4t = u4t = 2 Full capacity of cut is used

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**= Proof sketch of Max Flow=Min Cut S2 fs1+fs2=2+3=5**

2 2 4 3 2 2 3 0 2 1 1 0 t 1 1 1 0 s 2 2 4 3 2 4 2 4 = fs1+fs2=2+3=5 C(S2)=u13+u43+u4t= 2+1+2=5 from S2={s,1,2,4} to {3,t} f13 = u13 = 2 f43 = u43 = 1 f4t = u4t = 2 Full capacity of cut is used from {3,t} to {S2={s,1,2,4} f32 = 0 f34 = 0

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**= Proof sketch of Max Flow=Min Cut S2 fs1+fs2=2+3=5**

2 2 4 3 2 2 3 0 2 1 1 0 t 1 1 1 0 s 2 2 4 3 2 4 2 4 = fs1+fs2=2+3=5 C(S2)=u13+u43+u4t= 2+1+2=5 from S2={s,1,2,4} to {3,t} f13 = u13 = 2 f43 = u43 = 1 f4t = u4t = 2 Full capacity of cut is used from {3,t} to {S2={s,1,2,4} f32 = 0 f34 = 0 nothing flows back across the cut

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**= Proof sketch of Max Flow=Min Cut S2 fs1+fs2=2+3=5**

2 2 4 3 2 2 3 0 2 1 1 0 t 1 1 1 0 s 2 2 4 3 2 4 2 4 = fs1+fs2=2+3=5 C(S2)=u13+u43+u4t= 2+1+2=5 from S2={s,1,2,4} to {3,t} f13 = u13 = 2 f43 = u43 = 1 f4t = u4t = 2 from {3,t} to {S2={s,1,2,4} f32 = 0 f34 = 0 Capacity of the cut is equal to the flow from s to t

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