1

Ford-Fulkerson method

Ford-Fulkerson(G) f = 0

while(9 simple path p from s to t in Gf)

f := f + fp

output f

• Runs in time O(|fmax| |E|) where fmax is the maximum flow found.

2

Edmonds-Karp

• Edmonds-Karp: Implement Ford-Fulkerson by always choosing the shortest possible augmenting path, i.e., the one with fewest possible edges.

• Edmonds-Karp can be implemented to run in time O(|V| |E|2).

3

Maximum Matching

• Suppose we use the max flow formulation formulation of the maximum matching problem and solve the resulting flow instance using Edmonds-Karp.

• How fast an algorithm do we get?

4

Maximum Matching

• Answer 1: O(|V| |E|2).

• Answer 2: O(|E| |f|) = O(|V| |E|).

• Morale: Exponential time/polynomial time distinction removes a lot of information.

5

How do I get my (very big) max flow instance solved?

• Edmonds-Karp is not the most efficient known algorithm.

• Preflow-Push algorithms are better, both theoretically and in practice.

• Should I implement such algorithms myself?

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MaximumCardinalityMatching

Max Flow

Min Cost Flow

Alternating PathAlgorithm

Linear Programs

Reduction

Reduction

Reduction

Ford-Fulkerson,Edmonds-Karp

Augmenting cyclealgorithm

Simplex Algorithm,Interior point alg.

Algorithm

Algorithm

Algorithm

Incr

easi

ng g

ener

ality

Dec

reas

ing

effic

ienc

y

Algorithm

Reduction

AlgorithmMixed Int. Lin. Prog. Branch and Bound,Branch and Cut

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Standard state-of-the-art software

• ILOG CPLEX (www.ilog.com)

• X-press MP (www.dashoptimization.com)

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The Min Cost Flow Problem

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The Min Cost Flow problem

• We want to impose lower bounds as well as capacities on a given arc. Also, arcs should have costs.

• Rather than maximize the value (i.e. amount) of the flow through the network we want to minimize the cost of the flow.

10

Flow Networks with Costs

• Flow networks with costs are the problem instances of the min cost flow problem.

• A flow network with costs is given by 1) a directed graph G = (V,E) 2) capacities c: E ! R. 3) balances b: V ! R. 4) costs k: V £ V ! R. k(u,v) = -k(u,v).

• Convention: c(u,v)=0 for (u,v) not in E.

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Feasible Flows

• Given a flow network with costs, a feasible flow is a feasible solution to the min cost flow problem.

• A feasible flow is a map f: V £ V ! R satisfying capacity constraints: 8 (u,v): f(u,v) · c(u,v). Skew symmetry: 8 (u,v): f(u,v) = – f(v,u).

Balance Constraints: 8 u 2 V: v 2 V f(u,v) =b(u)

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Cost of Feasible Flows

• The cost of a feasible flow f is

cost(f)= ½ (u,v) 2 V £ V k(u,v) f(u,v)

• The Min Cost Flow Problem: Given a flow network with costs, find the feasible flow f that minimizes cost(f).

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Max Flow Problem vs. Min Cost Flow Problem

Max Flow Problem

Problem Instance:c: E ! R+.Special vertices s,t.

Feasible solution:8 u2V–{s,t}: v 2 V f(u,v) = 0

Objective:Maximize |f(s,V)|

Min Cost Flow Problem

Problem Instance:c: E ! R.Maps b,k.

Feasible solution:8 u2V: v 2 V f(u,v) = b(u)

Objective:Minimize cost(f)

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Negative Capacities

• c(u,v) < 0.

• Let l(v,u) = – c(u,v).

• f(u,v) · c(u,v) iff f(v,u) ¸ – c(u,v) = l(v,u).

• l(v,u) is a lower bound on the net flow from v to u for any feasible flow.

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Balance Constraints

• Vertices u with b(u)>0 are producing flow.

• Vertices u with b(u)<0 are consuming flow

• Vertices u with b(u)=0 are shipping flow.

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.

Can we solve the max-flow problem using software for the min-cost flow problem?

Unknown Balance!! Unknown Balance!!

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Capacity 29, Cost -1

All other costs 0, all balances 0.

Capacity 29, Cost 0

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Circulation networks

• Flow networks with b ´ 0 are called circulation networks.

• A feasible flow in a circulation network is called a feasible circulation.

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Unknown Balances!!

Unknown Balances!!

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21

Integrality theorem for min cost flow

If a flow network with costs has capacities and balances integer valued and a feasible flow in the network exists, then there is a minimum cost feasible flow which is integer valued.

(shown later by “type checking”)

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Minimum Weight Perfect Matching

• Given integer weight matrix

(w(i,j)), 1 · i,j · n.

• Find a permutation on {1,..,n} minimizing

i w(i,(i)).

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Tanker Scheduling Problem

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Capacity 1, Lower Bound 1, Cost 0

Capacity 1, Cost 0

Capacity 4, Cost 1All balances 0

Capacity 1, Cost 0

25

Hopping Airplane Problem

• Airplane must travel from city 1, to city 2, to city 3, .., to city n. At most p passengers can be carried at any time.

• bij passengers want to go from city i to city j and will pay fij for the trip.

• How many passengers should be picked up at each city in order to maximize profits?

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