hw3_sol_long - Com S 511 Homework#3 Due on Friday October 2...

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Com S 511: Homework #3 Due on Friday, October 2, 2009 Yuheng Long 1
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Yuheng Long Com S 511 : Homework #3 Problem 1 Problem 1 Problem 2 We will have the following constraints: The aggregate flow on an edge e must be no more than the capacity of the edge, c(e). f ( u, v ) = k i =1 f i ( u, v ) c ( u, v ) for each u, v V For an internal node, the amount of flow entering must equal the amount of flow leaving. v V f i ( u, v ) = 0 for each u V - { s i , t i } For a source(sink) s i ( t i ) node, the amount of flow leaving it must be exactly d i larger(smaller) than the amount of flow entering. v V f i ( s i , v ) = v V f i ( v, t i ) = d i We want to minimize the cost. Therefore, minimize u,v E ( a ( u, v ) k i =1 f i ( u, v )) subject to k i =1 f i ( u, v ) c ( u, v ) for each u, v V v V f i ( v, u ) = v V f i ( u, v ) for each i = 1 , 2 , . . . , k and for each u V - { s i , t i } v V f i ( s i , v ) = v V f i ( v, t i ) = d i for each i = 1 , 2 , . . . , k f i ( u, v ) 0 for each i = 1 , 2 , . . . , k and for each u, v V Problem 3 Consider the following linear program: maximize x - y subject to ax + by - 1 x, y 0 , (1) where a and b are real numbers.
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