Applications of the Max-Flow Min-Cut Theorem

S-T Cuts SF D H C A NY S = {SF, D, H}, T={C,A,NY} [S,T] = {(D,A),(D,C),(H,A)}, Cap [S,T] = = 11 S = {SF, D, A,C}, T = {H, NY} [S,T] = {(SF,H),(C,NY)}, Cap[S,T] =6 + 4 = 10 Partition the Nodes into Sets S and T. [S,T] = Arcs from Nodes in S to Nodes in T.

Maximum-Flow Minimum-Cut Theorem l Removing arcs (D,C) and (A,NY) cuts off SF from NY. l The set of arcs{(D,C), (A,NY)} is an s-t cut with capacity 2+7=9. l The value of a maximum s-t flow = the capacity of a minimum s-t cut. SF D H C A NY SF D H C A NY

Network Connectivity l An s,t vertex separator of a graph G=(V,E) is a set of vertices whose removal disconnects vertices s and t. l The s,t-connectivity of a graph G is the minimum size of an s,t vertex separator. l The vertex-connectivity of G is the min{s,t-connectivity of G: (s,t) in V}.

Example Some Vertex Separators For s=2, t=7: {3, 4, 6}, {1, 4, 8} 2,7-connectivity = 3 Some Vertex Separators For s=1, t=8: {4, 5, 6}, {2,3} 1,8-connectivity = 2 Vertex-Connectivity = 2

Menger’s Theorem l 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.

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

Proof l Lemma 1 –Each set of k vertex-disjoint s,t paths in G, corresponds to exactly one integral flow of value k in G’. l Lemma 2 –Each s,t cut of finite capacity c corresponds to an s,t vertex-separator of size c in G l Result follows from Max-Flow Min-Cut Theorem

Finding the Vertex-Connectivity of G=(N,A) l Let Node 1 be the Source Node s l Let c = |N| l For i = 2 … |N| –Let t = i –If s-t Connectivity < c Then Let c = s-t Connectivity l Return c Find Vertex-Connectivity of G with |N|-1 Maximum Flow Computations

Mining for Gold l Divide a cross-section of earth into a grid of equally-sized blocks. l 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. l 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.

Solution l 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). l 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. Don’t take block (3,1). * Assuming (1,3) Already Excavated

Maximum Flow Formulation l Each Block is a Node l Add Infinite Capacity Arc from Node (i,j) to Node (i-1,j-1), (i-1,j),(i-1,j+1) l Add Source Node s and Sink Node t l Wij = Vij - C Where Vij = Value of Gold in Block (i,j) C = Cost of Excavating a Block l If Wij < 0, Add an Arc from Node (i,j) to t with Capacity -Wij l If Wij > 0 Add an Arc from s to Node (i,j) with Capacity Wij

Maximum Flow Network 1,11,21,31,41,52,12,22,32,42,53,13,23,33,43,54,14,24,34,44,5st

Minimum Cut 1,11,21,31,41,52,12,22,32,42,53,13,23,33,43,54,14,24,34,44,5st Cut: S={s,(2,4)(1,3), (1,4), (1,5)} T: All Other Nodes

Meaning of the Minimum Cut l Excavate all Blocks in S –Only Finite-Capacity Arcs in Minimum Cut –Blocks in S Can be Excavated Without Excavating Blocks in T l 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 l Cap[S,T] = Value of Gold in T + Cost of Excavating S

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