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Set4 - Flow Decomposition and Transformations Flow...

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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 j 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 Arc Flows vs Path & Cycle Flows 1 2 4 5 3 6 2 3 4 10 3 4 4 7 5 9 2 -2 1 0 3 -4 i j b(i) b(j) x ij Cycles: Paths: 2-4-5-3-2: 3 units 1-2-4-5-6: 2 units 4-6-5-3-4: 3 units 5-3-4 : 1 unit 6-5-6: 2 units 2-4: 3 units 1-2-4-5-4-1 : 2 units 3-4-6-5-3: 3 units 6-5-6: 2 units
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5 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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6 Property of Cycle Flows If C is a cycle, then sending one unit of flow along C satisfies conservation of flow everywhere. 1 2 3 4 5 1 1 1 1 1 C
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7 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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8 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. Flow Decomposition Ani
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9 Flow Decomposition Notation: G = (N, A) : the network x : Initial flow y : Flow at intermediate stage of the algorithm A(y) arcs with positive flow in y N(y) nodes incident to an arc in A(y) G(y) intermediate network (arcs with zero y values eliminated) SupplyNodes(y) nodes with supply wrt y DemandNodes(y) nodes with demand wrt y P : paths with flow in the decomposition C : cycles with flow in the decomposition
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10 Flow Decomposition Algorithm Initialize begin y := x; P := ; C := ; end Select(s, y) begin if SupplyNodes(y) , then s SupplyNodes(y); else s N(y) end Search(s, y) Carry out a depth first search starting with node s until finding a directed cycle C in G(y) or a path P in G(y) from s to a node t in DemandNodes(y).
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11 Capacities of Paths and Cycles s 4 2 9 t 6 4 3 5 P 1 4 2 7 5 5 4 7 9 8 C The capacity of C is denoted as (C, y) = min arc flow on C wrt flow y.
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