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# (Solved): Problem 1 Setup Sometimes, in a linear program, we need to convert constraints from one form to anot ...

Problem 1 Setup Sometimes, in a linear program, we need to convert constraints from one form to another. Part A Show how to convert an equality constraint into an equivalent set of inequalities. That is, given a constraint

`|\$\Sigma _(j)`

|

`=1^(n)`

a_{ij}x_j = b_i

`|`

| \$, give a set of inequalities that will be satisfied if an only if

`|\$\Sigma _(j)`

|

`=1^(^()){n}`

a_{ij}x_j = b_i|\$. Part B Show how to convert an inequality constraint

`|\$\Sigma _(j)`

|

`=1^(^()){n}`

a_{ij}x_j <= b_i

`1\$`

into an equality constraint and a nonnegativity constraint. You will need to introduce an additional variable

`(/)/(\$)\$\$`

, and use the constraint that

`|\$s:`

|. Problem 2 Rewrite the linear program for maximum flow (from Chapter 29.2) so that it uses only

`|\$O(V+E)|\$`

constraints. Problem 3 Show that the dual of the dual of a linear program is the primal linear program. Problem 4 Show that if an edge

`(/)/(\$)(u,v)|\$`

| is contained in some minimum spanning tree, then it is a light edge corssing some cut of the graph. Problem 2 Show how to find a maximum flow in a flow network

`(/)/(\$)G=(V,E)|\$`

| by a sequence of at most

`|\$|E||\$`

augmenting paths. Hint Determine the paths after finding the maximum flow.

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