max-flow-applications-4up - Princeton University • COS...

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Unformatted text preview: Princeton University • COS 423 • Theory of Algorithms • Spring 2001 • Kevin Wayne Maximum Flow Applications 2 Maximum Flow Applications Contents Max flow extensions and applications. ■ Disjoint paths and network connectivity. ■ Bipartite matchings. ■ Circulations with upper and lower bounds. ■ Census tabulation (matrix rounding). ■ Airline scheduling. ■ Image segmentation. ■ Project selection (max weight closure). ■ Baseball elimination. 3 Disjoint path network: G = (V, E, s, t). ■ Directed graph (V, E), source s, sink t. ■ Two paths are edge-disjoint if they have no arc in common. Disjoint path problem: find max number of edge-disjoint s-t paths. ■ Application: communication networks. s 2 3 4 Disjoint Paths 5 6 7 t 4 Disjoint path network: G = (V, E, s, t). ■ Directed graph (V, E), source s, sink t. ■ Two paths are edge-disjoint if they have no arc in common. Disjoint path problem: find max number of edge-disjoint s-t paths. s 2 3 4 Disjoint Paths 5 6 7 t 5 Max flow formulation: assign unit capacity to every edge. Theorem. There are k edge-disjoint paths from s to t if and only if the max flow value is k. Proof. ⇒ ■ Suppose there are k edge-disjoint paths P 1 , . . . , P k . ■ Set f(e) = 1 if e participates in some path P i ; otherwise, set f(e) = 0. ■ Since paths are edge-disjoint, f is a flow of value k. Disjoint Paths s t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 Max flow formulation: assign unit capacity to every edge. Theorem. There are k edge-disjoint paths from s to t if and only if the max flow value is k. Proof. ⇐ ■ Suppose max flow value is k. By integrality theorem, there exists {0, 1} flow f of value k. ■ Consider edge (s,v) with f(s,v) = 1. – by conservation, there exists an arc (v,w) with f(v,w) = 1 – continue until reach t, always choosing a new edge ■ Produces k (not necessarily simple) edge-disjoint paths. Disjoint Paths s t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 Network connectivity network: G = (V, E, s, t) . ■ Directed graph (V, E), source s, sink t. ■ A set of edges F ⊆ E disconnects t from s if all s-t paths uses at least on edge in F. Network connectivity: find min number of edges whose removal disconnects t from s. Network Connectivity s 2 3 4 5 6 7 t 8 Network connectivity network: G = (V, E, s, t) . ■ Directed graph (V, E), source s, sink t. ■ A set of edges F ⊆ E disconnects t from s if all s-t paths uses at least on edge in F. Network connectivity: find min number of edges whose removal disconnects t from s. Network Connectivity s 2 3 4 5 6 7 t 9 Disjoint Paths and Network Connectivity Menger’s Theorem (1927). The max number of edge-disjoint s-t paths is equal to the min number of arcs whose removal disconnects t from s....
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This note was uploaded on 04/15/2010 for the course INDUSTRIAL ie513 taught by Professor Zeynephuygur during the Spring '10 term at Bilkent University.

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max-flow-applications-4up - Princeton University • COS...

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