Linear Programming Max Flow Min Cut

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Linear Programming Max Flow Min Cut

• Orgad Keller
• Recitation 6

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Linear Programming

• We have an objective function
• Subject to constraints

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Linear Programming

• Given all parameters
• we want to find the optimal

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Linear Programming

• It is easier to present the problem with a matrix
  and vectors

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The Dual Problem

• Given the Primal Problem
• Its Dual Problem is defined as

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Strong Duality Theorem

• Given a problem and its dual problem, then
• In other words, the optimal objective functions
  value for the primal problem, is equal to the
  optimal objective functions value for the dual
  problem.

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Maximum Flow

• As you remember from Algorithms I
• Given a directed graph , two
  vertices and a capacity for each
  edge
• We want to find a flow function
• so that the flow value is
  maximal

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Maximum Flow

• But, we are subject to some rules
• What goes in must come out
• Capacity restrictions

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Max Flow with Linear Programming

• Well show by a simple example

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Max Flow with Linear Programming

• We want to present the example in the form
• Subject to

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Max Flow with Linear Programming

• Formally

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Max Flow with Linear Programming

• So well add another edge, and change the
  problems representation a little.

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Minimum Cut

• As you remember from Algorithms I
• Given a directed graph , two
  vertices and a weight for each edge
• We want to find a minimal-weight subset of edges
  such that if well remove them, we wont be able
  to travel from to .

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Minimum Cut

• In other words
• Well choose ,
• where , ,
• such that the cut value,
• is minimal.

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Min Cut with Linear Programming

• Going back to the example

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Min Cut with Linear Programming

• What about
• Is that enough?
• We havent ensured paths from to are cut.

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Min Cut with Linear Programming

• Beside a variable for every edge ,
  well want a variable for every vertex
  , such that

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Min Cut with Linear Programming

• Lets take for instance
• If is in the cut ,
  that means that
• If is not in the cut , that
  means that either or
  or
• So it is the same to constrain

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Max Flow with Linear Programming

• Formally

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Max Flow Min Cut Theorem

• We now see that the problems are dual
• So also according to the Strong Duality Theorem,
  Max Flow Min Cut

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Integer Values

• We want integer values for the variables, but
  this is LP, not IP!
• So how do we know that the solution will yield
  integer values for the variables?
• Theroem If the constraints matrix is totally
  unimodular and the right hand side is comprised
  of integers, then its easy to find an integer
  solution.

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Unimodular / Totally Unimodular

• Definition A Unimodular matrix is a square
  matrix whose determinant is 0, 1 or -1.
• Definition A Totally Unimodular matrix is a
  matrix whose every non-singular square submatrix
  is unimodular.