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Unformatted text preview: MaxFlow Problems MaxFlow is a graph problem that seems very specific and narrowly defined. But many seemingly unrelated problems can be converted to maxflow problems. A flow network consists of: A directed graph G = ( V , E ) . Each edge u → v ∈ E has a capacity c ( u , v ) ≥ . (If u → v ∈ E then c ( u , v ) = .) Two special vertices: the source s and the sink t . For each vertex v ∈ V , there is a directed path s → t passing through v . Note: The last condition is not essential. It is included here for convenience. c Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 2 / 76 MaxFlow: Definitions Flow Function A flow is a real valued function f : V × V → R that satisfies the following conditions: Capacity Constraint: For all u , v ∈ V , f ( u , v ) ≤ c ( u , v ) . Skew Symmetry Constraint: For all u , v ∈ V , f ( v , u ) = f ( u , v ) . Flow Conservation Constraint: For any u ∈ V{ s , t } , v ∈ V f ( u , v ) = The flow value of f is defined to be:  f  = v ∈ V f ( s , v ) f ( u , v ) is called the net flow from u to v . c Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 3 / 76 MaxFlow: Example t v1 v2 v3 v4 11/16 8/13 12/12 15/20 11/14 4/9 7/7 4/4 1/41/10 s In this figure, the notation 11 / 16 means f ( s , v 1 ) = 11 and c ( s , v 1 ) = 16 . The edges with 0 capacity are not shown. Only positive flow values are shown. (Recall that f ( v , u ) = f ( u , v ) by the skew symmetry constraint.) The capacity constraint is satisfied at all edges. The conservation constraint at the vertex v 1 is: f ( v 1 , s ) + f ( v 1 , v 2 ) + f ( v 1 , v 3 ) + f ( v 1 , v 4 ) + f ( v 1 , t ) = 11 + 1 + 12 + + = The flow value is:  f  = 11 + 8 = 19 . c Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 4 / 76 MaxFlow: Definition Caution In the example above, suppose that 3 units flow v 2 → v 1 , and 2 units flow v 1 → v 2 . It can be seen that the flow conservation and the capacity constrains are still satisfied. But are f ( v 2 → v 1 ) = 3 and f ( v 2 → v 1 ) = 2 ? Then the Skew Symmetry constrains f ( v 1 , v 2 ) = f ( v 2 , v 1 ) would not be satisfied. In the formulation of the maxflow problem, shipping 3 units flow from v 2 to v 1 , and 2 units flow from v 1 to v 2 is equivalent to shipping 1 unit flow from v 2 to v 1 , and nothing from v 1 to v 2 . In other words, the 2 units flow from v 2 to v 1 , then from v 1 back to v 2 are canceled . So we have: f ( v 2 , v 1 ) = 1 and f ( v 1 , v 2 ) = 1 c Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 5 / 76 MaxFlow: Application G = ( V , E ) represents an oil pipeline system Each edge u → v ∈ E is an onedirectional pipeline....
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 Fall '11
 XINHE
 Algorithms, Flow network, Maximum flow problem, Maxflow mincut theorem, Network flow

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