Modelling with Max Flow

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? 3

M=3 4

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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 FordFulkerson method will yield such a maximum flow. The integrality theorem is often extremely important when “programming” and modeling using the max flow formalism. 7

Reduction: Maximum Matching → Max Flow

What is the maximum cardinality matching in G? 8

G 9

s

t

G’

All capacities are 1 10

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 11

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. 13

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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 TwoProcessor 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. 16

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

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König’s 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. 22