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.