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**Unformatted text preview: **Max-Flow Problems Max-Flow is a graph problem that seems very specific and narrowly defined. But many seemingly unrelated problems can be converted to max-flow 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 Max-Flow: 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 Max-Flow: Example t v1 v2 v3 v4 11/16 8/13 12/12 15/20 11/14 4/9 7/7 4/4 1/4-1/10 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 Max-Flow: 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 max-flow 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 Max-Flow: Application G = ( V , E ) represents an oil pipeline system Each edge u v E is an one-directional pipeline....

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