04transformationsandflowdecomposition

04transformationsandflowdecomposition - 15.082 and 6.855J...

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1 15.082 and 6.855J February 13, 2003 Flow Decomposition and Transformations
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2 Flow Decomposition and Transformations ¡ Flow Decomposition ¡ Removing Lower Bounds ¡ Removing Upper Bounds ¡ Node splitting ¡ Arc flows : an arc flow x is a vector x satisfying: Let b(i) = 6 j x ij - 6 i x ji We are not focused on upper and lower bounds on x for now.
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3 Flows along Paths Usual : represent flows in terms of flows in arcs. Alternative : represent a flow as the sum of flows in paths and cycles. Two units of flow in the path P 1 2 3 4 5 2 2 2 2 P 1 2 3 4 5 1 1 1 1 1 C One unit of flow around the cycle C
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4 Properties of Path Flows Let P be a directed path. Let Flow( G ,P) be a flow of G units in each arc of the path P. 1 2 3 4 5 2 2 2 2 P Flow(2, P) Observation. If P is a path from s to t, then Flow( G ,P) sends G# units of flow from s to t, and has conservation of flow at other nodes.
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5 Property of Cycle Flows ¡ If p is a cycle, then sending one unit of flow along p satisfies conservation of flow everywhere. 1 2 3 4 5 1 1 1 1 1
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6 Representations as Flows along Paths and Cycles Let P be a collection of Paths; let f(P) denote the flow in path P G ij (P) = 1 if (i,j) ¢ P G ij (P) = 0 if (i,j) £ P Let C be a collection of cycles; let f(C) denote the flow in cycle C. G ij (C) = 1 if (i,j) ¢ C G ij (C) = 0 if (i,j) £ C
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7 Representations as Flows along Paths and Cycles ¡ Claim: one can convert the path and cycle flows into an arc flow x as follows: for each arc (i,j) ¢ A x ij = 6 P ¢ P G ij (P)f(P) + 6 C ¢ C G ij (C)f(C) ¡ We next provide an algorithm for converting arc flows to sums of flows around cycles and along paths, where each path is from a supply node wrt x to a demand node wrt x.
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This note was uploaded on 03/15/2010 for the course IE 505 taught by Professor Yok during the Spring '10 term at Galatasaray Üniversitesi.

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04transformationsandflowdecomposition - 15.082 and 6.855J...

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