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 $|$$${}_j|=1^n$ a$\_\{$ij$\}$x$\_$j $=$ b$\_$i $||$ $$,$ give a set of inequalities that will be satisfied if an only if $|$$${}_j|=1^{{?}^{?}}\{n\}$ a$\_\{$ij$\}$x$\_$j $=$ b$\_$i$|$$$.$

Part B

Show how to convert an inequality constraint $|$$${}_j|=1^{{?}^{?}}\{n\}$ a$\_\{$ij$\}$x$\_$j b$\_$i $1$$ into an equality constraint and a nonnegativity constraint. You will need to introduce an additional variable $\frac{\frac{?}{?}}{\text{\$}}$$$$,$ and use the constraint that $|$$$s$:$|.$

Problem $2$

Rewrite the linear program for maximum flow $($from Chapter $29\mathrm{.}2)$ 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 $\frac{\frac{?}{?}}{\text{\$}}(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 $\frac{\frac{?}{?}}{\text{\$}}G=(V,E)|$$$|$ by a sequence of at most $|$$$|E||$$ augmenting paths.

Hint

Determine the paths after finding the maximum flow.