04_Flow_Decomposition

04_Flow_Decomposition - 15.082 and 6.855J February 13, 2003...

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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) = Σ j x ij - Σ 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 One unit of flow around the cycle C 1 2 3 4 5 2 2 2 2 P 1 2 3 4 5 1 1 1 1 1 C
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4 Properties of Path Flows Let P be a directed path. Let Flow( δ ,P) be a flow of δ units in each arc of the path P. Observation. If P is a path from s to t, then Flow( δ ,P) sends δ units of flow from s to t, and has conservation of flow at other nodes. 1 2 3 4 5 2 2 2 2 P Flow(2, P)
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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 C be a collection of cycles; let f(C) denote the flow in cycle C. δ ij (C) = 1 if (i,j) C δ ij (C) = 0 if (i,j) C Let P be a collection of Paths; let f(P) denote the flow in path P δ ij (P) = 1 if (i,j) P δ ij (P) = 0 if (i,j) P
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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 = Σ P P δ ij (P)f(P) + Σ C C δ 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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04_Flow_Decomposition - 15.082 and 6.855J February 13, 2003...

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