Reading before Lecture 13 Network Flow

Flow Decomposition and Transformations

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The Flow Decomposition Algorithm Obtain a network, and use the same network to illustrate the shortest path problem for communication networks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.)

begin Initialize while y   do Select(s, y) Search(s, y) if a cycle C is found then do let D = Capacity(C, y) Add Flow(D, C) to cycle flows Subtract Flow(D, C) from y. end if a path P is found then do let D = Capacity(P, y) Add Flow(D, P) to path flows Subtract Flow(D, P) from y.

The initial flow 7 2 4 4 3 3 8 2 1 6 -2 4 4 2 3 5 5 9 A supply node The flow x A demand node A balanced node

Find a Path or Cycle 7 2 4 4 3 3 8 2 1 1 6 -2 4 4 Carry out a depth first search until a cycle C or a path P is found Select a supply node 2 3 5 5 9 Determine the capacity The capacity is 2.

Update the decomposition 5 7 2 4 2 4 3 3 8 6 2 1 1 6 -2 4 4 Add the cycle flow to the decomposition 2 3 5 5 9 7 update the current flow 2 units around 1-2-4-5-3-1 cycle flows path flows

Find the next path or cycle 5 2 4 3 2 3 6 Start at a supply node and find the next cycle or path 2 1 1 6 -2 4 4 3 5 5 7 2 units around 1-2-4-5-3-1 cycle flows path flows

update the decomposition and current flow 2 5 2 4 3 2 3 6 3 update the current flow and decomposition 2 1 1 6 -2 4 4 3 5 5 7 4 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 cycle flows path flows

Find the next flow or cycle 2 2 4 3 2 start with a supply node and find the next flow or cycle. 3 2 1 1 6 -2 4 4 3 5 5 4 The capacity of 4-5-3-4 is 3 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 cycle flows path flows

update the current flow and decomposition 2 1 1 6 -2 4 1 4 3 2 3 update the current flow and decomposition 2 1 1 6 -2 4 1 4 3 5 5 1 4 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 3 units around 4-5-3-4 cycle flows path flows

Find the next flow or cycle start with a supply node and find the next flow or cycle. 2 2 4 3 2 2 1 1 6 -2 1 4 The capacity of 1-2-4-6 is 2 3 5 5 1 4 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 3 units around 4-5-3-4 cycle flows path flows

update the current flow and decomposition 2 2 4 3 1 2 2 2 4 3 1 2 2 1 1 6 -2 1 4 3 5 5 1 4 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 2 units in 1-2-4-6 3 units around 4-5-3-4 cycle flows path flows

Find the next path or cycle start with any node incident to an arc with flow and find the next flow or cycle. 2 4 1 1 6 -2 1 4 3 3 5 5 4 1 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 2 units in 1-2-4-6 3 units around 4-5-3-4 cycle flows path flows

Update 2 4 1 1 6 -2 1 4 3 3 5 5 4 4 1 cycle flows path flows 2 4 1 1 6 -2 1 4 3 3 5 5 4 4 1 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 2 units in 1-2-4-6 3 units around 4-5-3-4 1 units around 3-4-6-5-3 cycle flows path flows

The Final Decomposition 2 4 1 6 -2 4 3 5 4 2 units around 1-2-4-5-3-1 3 units around 2-4-5-3-2 3 units around 4-5-3-4 2 units in 1-2-4-6 1 units around 3-4-6-5-3 4 units around 5-6-5 cycle flows path flows

(yki + lki)

bi 1 1 bj+uij -uij -1 -1 -uij -1 -1 -uij xij xij sij bi bj+uij bj bi

Exercise Transform the following network for removing arc capacities. (5,10) (6,30) (cij, uij) (4,20) (1,10)

Homework Readings Section 2.4 Section 3.5 Exercises Exercise 2.42