# 6417c11 - Maximum Flows CONTENTS Introduction to Maximum...

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1 Maximum Flows CONTENTS Introduction to Maximum Flows (Section 6.1) Introduction to Minimum Cuts (Section 6.1) Applications of Maximum Flows (Section 6.2) Flows and Cuts (Section 6.3) Network Simplex Algorithm (Section 11.8) Generic Augmenting Path Algorithm (Section 6.4) Max-Flow Min-Cut Theorem (Section 6.5) Capacity Scaling Algorithm (Section 7.3) Generic Preflow-Push Algorithm (Section 7.6) Specific Preflow-Push Algorithms (Section 7.8)

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2 Problem Definition Maximum Flow Problem : Given a directed network G with arc capacities given by u ij ’s, determine the maximum amount of flow that can be sent from a source node s to a sink node t. Decision Variables : Flow x ij on arc (i, j), and the flow v entering the sink node. 1 2 4 3 5 6 10 25 35 20 15 35 40 30 20 s t
3 A Linear Programming Problem The maximum flow problem can be formulated as the following linear programming problem: Maximize v subject to x ij 0 for every arc (i, j) A x x for i s, 0 for i s, t for ij ji {j:(j,i) A} {j:(i,j) A} - = = - = R S | T | v v i t,

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4 An Alternate Formulation The maximum flow problem can also be conceived of as the following minimum cost flow problem: Minimize -x ts subject to x ij 0 for every arc (i, j) A x x ij ji {j:(j,i) A} {j:(i,j) A} - = 0 20 1 2 4 3 5 6 10 25 35 15 35 40 30 Cost = -1, Capacity = s t 25
5 Assumptions All arc costs (or lengths) c ij 's are integer and C is the largest magnitudes of arc costs. Rational arc costs can be converted to integer arc costs. Irrational arc costs (such as, ) cannot be handled. The network contains a directed path from node s to every node in the network. We can add artificial arcs of large cost to satisfy this assumption. The network does not contain a negative cycle. In the presence of negative cycles, the optimal solution of the shortest path problem is unbounded. The network is directed. To satisfy this assumption, replace each undirected arc (i, j) with cost c ij by two directed arc (i, j) and (j, i) with cost c ij . 2

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6 Minimum Cut Problem A cut is a (minimal) set of arcs whose deletion from G disconnects the network (into two parts S and ). An s-t cut is a cut that disconnects nodes s and t. We denote an s-t cut as [S, ]. 1 2 4 3 5 6 10 25 35 20 15 35 40 30 20 s t S S
7 Minimum Cut Problem (contd.) (S, ): Set of forward arcs in the cut [S, ] ( , S): Set of backward arcs in the cut [S, ] u[S, ] : Capacity of the cut [S, ] defined as S 1 2 4 3 5 6 10 25 35 20 15 35 40 30 20 s t S S S S S u S S u ij i j S,S [ , ] ( , ) ( ) =

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8 Assumptions 1. Network is directed. How to satisfy this condition? 2. All arc capacities are nonnegative integers. Let U denote the largest arc capacity. 3. The network does not contain an uncapacitated (that is, infinite capacity) directed path from node s to node t. How to identify such a possibility?
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• Spring '07
• SIRIPHONGLAWPHONGPANICH
• Shortest path problem, Flow network, Maximum flow problem, Max-flow min-cut theorem, maximum flow

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