Modelling with Max Flow

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The Max Flow Problem

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Modeling with Max Flow:A 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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G’

s t

All capacities are 1

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

• 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.

(Ahuja, Application 6.2)

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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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• 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.

Distributed Computation on Two-Processor Computer

(Ahuja, Application 6.5)

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

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Find a subset of regions to mine so that the total profit is maximized.

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.

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