Modelling with Max Flow

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Modelling with Max Flow
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The Max Flow Problem
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Modeling with Max FlowA scheduling problem

• A set of jobs must be scheduled on M identical
  machines.
• Each job j has an release (arrival) date rj, a
  required due date dj and a processing time pj
  dj - rj.
• A job can be preemptively moved from one machine
  to another.
• Can the jobs be scheduled on the machines so that
  the deadlines are met?

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M 3
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Basic property of model

• Feasible (legal) schedules correspond to flows
  that saturate all outgoing arcs of s.
• correspond to time spent on a particular job
  on a particular set of dates can be read off from
  flow along arcs in middle layer.

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Integrality Theorem (26.11)

• If a flow network has integer valued
  capacities, there is a maximum flow with an
  integer value on every edge. The Ford-Fulkerson
  method will yield such a maximum flow.
• The integrality theorem is often extremely
  important when programming and modeling using
  the max flow formalism.

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Reduction Maximum Matching ! Max Flow
What is the maximum cardinality matching in G?
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G
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s
t
G
All capacities are 1
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Relating G and G

• Matchings in G correspond exactly to integral
  flows of G
• Correspondence
• Arcs with a flow of 1 correspond to edges in the
  matching.
• Arcs with a flow of 0 correspond to non-edges
• A max flow which is integral correspond to a
  maximum matching

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Integrality essential
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Finding a balanced set of Representatives
(Ahuja, Application 6.2)

• A city has clubs C1, C2,,Cn and parties P1,
  P2,,Pm. A citizen may be a member of several
  clubs but may only be a member of one party.
• A balanced city council must be formed by
  including exactly one member from each club and
  at most uk members from party Pk.

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Max Flow Min Cut Theorem

• The value of the maximum flow in G is equal to
  the capacity of the minimum cut in G.

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Distributed Computation on Two-Processor Computer
(Ahuja, Application 6.5)

• Processes p1, p2, , pn must be assigned to one
  of two processors.
• Assigning pi to processor k gives computation
  cost aik.
• If pi and pk are assigned to different
  processors, communication cost cik is incurred.
• Minimize the total cost.

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but there is a lot of power of in modeling with
directed cuts
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Königs theorem

• The size of the largest matching in a bipartite
  graph is equal to the size of the smallest vertex
  cover.

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Find a subset of regions to mine so that the
total profit is maximized.
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When solving exam problems

• Flow networks is a graphical formalism. This does
  not mean that a sloppy drawing is sufficient to
  specify a model.
• . remember that max flow networks are directed
  graphs.
• .. remember that arcs in a max flow network have
  capacities that much be specified.