duality and the max flow min-cut theorem in linear optimization

duality and the max flow min-cut theorem in linear optimization

duality and the max flow min-cut theorem in linear optimization

Instructions:

the problem is in the Introduction to Linear Optimization Dimitris Bertsimas John N. Tsitsiklis text book at the page351 exercise 7.20, I am writing the problem below:

consider the maximumflow problem.
(a) Let pi be a price variable associated with the flow conservation constraint at node i . Let qij be aprice variable associated with the capacity constraint at arc (i,j). Write down a minimization problem, with variables pi and qij, whose dual is the maximum flow problem.
(b)show that the optimal value in the minimization problem is equal to the minimum cut capacity, and prove the max-flow min-cut theorem.