Applications of the Max-Flow Min-Cut Theorem

1
Applications of the Max-Flow Min-Cut Theorem
2
S-T Cuts
Partition the Nodes into Sets S and T. S,T
Arcs from Nodes in S to Nodes in T.
S SF, D, H, TC,A,NY S,T
(D,A),(D,C),(H,A), Cap S,T 245 11 S
SF, D, A,C, T H, NY S,T
(SF,H),(C,NY), CapS,T 6 4 10
3
Maximum-Flow Minimum-Cut Theorem

• Removing arcs (D,C) and (A,NY) cuts off SF from
  NY.
• The set of arcs(D,C), (A,NY) is an s-t cut with
  capacity 279.
• The value of a maximum s-t flow the capacity of
  a minimum s-t cut.

4
Network Connectivity

• An s,t vertex separator of a graph G(V,E) is a
  set of vertices whose removal disconnects
  vertices s and t.
• The s,t-connectivity of a graph G is the minimum
  size of an s,t vertex separator.
• The vertex-connectivity of G is the
  mins,t-connectivity of G (s,t) in V.

5
Example
6
2
1
8
4
3
7
5
Some Vertex Separators For s2, t7 3, 4, 6,
1, 4, 8 2,7-connectivity 3 Some Vertex
Separators For s1, t8 4, 5, 6,
2,3 1,8-connectivity 2 Vertex-Connectivity
2
6
Mengers Theorem

• Given and undirected graph G and two nonadjacent
  vertices s and t, the maximum number of
  vertex-disjoint (aside from sharing s and t)
  paths from s to t is equal to the
  s,t-connectivity of G.

7
Maximum Flow Formulation

• For G(V,E), construct the network G(N,A) as
  follows
• For each vertex v in V/s,t, add nodes v and v
• Add arc (v,v) with capacity 1
• For each edge (u,v) in E, add arcs (u,v) and
  (v,u) with infinite capacity
• For each edge (s,v) in E, add arc (s,v) with
  infinite capacity
• For each edge (v,t) in E, add arc (v,t) with
  infinite capacity

8
Proof

• Lemma 1
• Each set of k vertex-disjoint s,t paths in G,
  corresponds to exactly one integral flow of value
  k in G.
• Lemma 2
• Each s,t cut of finite capacity c corresponds to
  an s,t vertex-separator of size c in G
• Result follows from Max-Flow Min-Cut Theorem

9
Finding the Vertex-Connectivity of G(N,A)

• Let Node 1 be the Source Node s
• Let c N
• For i 2 N
• Let t i
• If s-t Connectivity lt c Then
• Let c s-t Connectivity
• Return c
• Find Vertex-Connectivity of G with N-1 Maximum
  Flow Computations

10
Mining for Gold

• Divide a cross-section of earth into a grid of
  equally-sized blocks.
• Given the cost of excavating a full block and the
  market value of a full block of gold, determine
  an optimal shape for the mine.
• Constraints on the shape of your mine require
  that a block of earth cannot be excavated unless
  the three blocks above it (directly, and
  diagonally left and right) have also been
  excavated.

11
(No Transcript)
12
Solution

• To get the gold in block (2,4), we need to
  excavate blocks (1,3), (1,4), (1,5) and (2,4).
  Revenue 500, Cost 400, Profit 100. Take
  block (2,4).
• To get the gold in block (3,1), we need to
  excavate blocks (3,1), (2,1), (2,2), (1,1), (1,2)
  and (1,3). Revenue 500, Cost 500, Profit
  0. Dont take block (3,1).
• Assuming (1,3) Already Excavated

13
Maximum Flow Formulation

• Each Block is a Node
• Add Infinite Capacity Arc from Node (i,j) to Node
  (i-1,j-1), (i-1,j),(i-1,j1)
• Add Source Node s and Sink Node t
• Wij Vij - C Where
• Vij Value of Gold in Block (i,j)
• C Cost of Excavating a Block
• If Wij lt 0, Add an Arc from Node (i,j) to t with
  Capacity -Wij
• If Wij gt 0 Add an Arc from s to Node (i,j) with
  Capacity Wij

14
Maximum Flow Network
100
100
100
100
100
400
400
15
Minimum Cut
Cut Ss,(2,4)(1,3), (1,4), (1,5) T All Other
Nodes
100
100
100
100
100
400
400
16
Meaning of the Minimum Cut

• Excavate all Blocks in S
• Only Finite-Capacity Arcs in Minimum Cut
• Blocks in S Can be Excavated Without Excavating
  Blocks in T
• Arcs in S,T
• Arcs from s to Nodes in T
• Capacity Value of Gold in T
• Arcs from Nodes in S to t
• Capacity Cost of Excavating S
• CapS,T
• Value of Gold in T Cost of Excavating S

17
Maximizing Profit

• Finding a Maximum Flow Minimizes
• Value of Gold in T Cost of Excavating S
• Equivalent to Maximizing
• F - (Value of Gold in T Cost of Excavating S
  )
• Equivalent to Maximizing
• Value of All Gold F
• Value of Gold in S - Cost of Excavating S
• Profit for Excavating S