Consider the following input for the minimum-congestion multicommodity flow problem: the flow network as given below and 2 commodities (s1, t1, 40) and (s2, t2, 80).The current simultaneous flow sends 30 units of commodity 1 along path (s1, a, b, t1) , 10 units of commodity 1 along path (s1, c, e, t1), 40 units of commodity 2 along path (s2, c, e, t2) and 40 units of commodity 2 along path (s2, p, q, t2).Complete the following statements by dragging and dropping appropriate options.

For a minimum-cost flow problem to have a feasible solution, which of the following mustbe true?A) There is the same number of supply nodes and demand nodes.B) There is only one supply node and one demand node.C) There is an equal amount of supply and demand.D) The supply and demand must be integers.E) The transshipment nodes must be able to absorb flow.

The diagram below shows an input network for the minimum-cost flow problem, and the flow (shown in red) in this network at the end of the 2-nd iteration of the Successive Shortest Path algorithm. During the computation, whenever there is a choice of a vertex, the algorithm selects the vertex which is first in the lexicographical order. (This means, for example, that during the first iteration, the algorithm selected vertex p1 from the two available supply vertices, and selected vertex r1 from the two available demand vertices.) Complete the following statements by dragging and dropping appropriate options.There was  Blank 1 Question 1 path from p2 to r1 in the residual network at the beginning of the 2-nd iteration.At the end of the 2-nd iteration, the flow saturates  Blank 2 Question 1 edges.At the beginning of the 3-rd iteration, the residual supply at vertex p2 is  Blank 3 Question 1 .In the residual network constructed in the 3-rd iteration,  Blank 4 Question 1 vertices are reachable from vertex p2.The cost of the path selected in the residual network in the 3-rd iteration is  Blank 5 Question 1 .The flow in the input network computed by the end of the 3-rd iteration  Blank 6 Question 1 satisfy all supply and demand.

Con-sider the uncapacitated network flow problem.(a) Show that every spanning tree determines a basic solution to the dualproblem

A minimum-cost flow problem is a special type of:A) linear programming problem.B) transportation problem.C) spanning tree problem.D) transshipment problem.E) maximum flow problem.

Which of the following will have negative net flow in a minimum-cost flow problem?A) Supply nodesB) Transshipment nodesC) Demand nodesD) Arc capacitiesE) None of the choices is correct.