(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.
