Consider the directed network (D, c) given by the following drawing, where each arc e E A(D) is labelled by its capacity c(e) and two vertices s and t have been identified. 5 2 2 2 6 2 3 4 3 6 Use the Ford-Fulkerson algorithm to find a maximum s-t-flow of (D, c). Draw the residual network after each iteration of the algorithm and give the size of the maximum flow.

Let V = {V₁, V2, V3} and R = {(V₁, V₁), (V₁, V2), (V₁, V3), (V2, V2), (V2, V3), (V3, V₁)}. a. Draw a directed graph to represent R and find MR. b. Find So, S₁, S2, and the shortest path matrix, S3.

A B UD C D E F A B c|1775| 12 39 39 17 25 с 25 E F 23 31 40 4 20 1 7 35 D 12 15 18 32 23 4 31 20 7 15 40 1 35 18 32 -- The weights of edges in a graph are shown in the table above. Find the minimum cost spanning tree on the graph above using Kruskal's algorithm. What is the total cost of the tree?

We consider the Traveling Salesman Problem with the cost matrix 0 87 1 5 1086 3 7 10 15 9 2 70 5 8 2620 and apply Little's branch and bound algorithm. What is the initial lower bound on possible solutions at the very first step of the algorithm? Which edge (which entry of the cost matrix) is used for the first inclusion/exclusion branching? Enter the indices of vertices at both end points of this edge without any separators between them. (For example,if the edge (12) (or,in other words,the matrix entry c12) is chosen for inclusion/exclusion, enter 12 in the text box.) What is the lower bound on the solutions in the exclusion branch (after the fırst inclusion/exclusion branching)?

A King in ancient times agreed to reward the inventor of chess with one grain of wheat on the first of the 64 squares of a chessboard. On the second square, the King would place two grains of​ wheat, on the third​ square, four grains of​ wheat, and on the fourth square eight grains of wheat. If the amount of wheat is doubled in this way on each of the remaining​ squares, how many grains of wheat should be placed on a square 16​? Also, find the total number of grains of wheat on the board at this time and their total weight in pounds.​ (Assume that each grain of wheat weighs​ 1/7000 pounds.)

4. (a) Use Dijkstra's algorithm to find the shortest path from node A to node F in the following network and give its length. A с 2 5 B 24 13 6 3 F D 14 Weight [kg] 40 50 60 40 250 200 70 190 1 E (b) A museum wants to ship 8 Roman statues for a special exhibition. They are to be transported in crates each holding a maximum weight of 300 kilograms. The numbers in the table represent the weight in kilograms of the statues: Statue Mercury Venus Tellus Mars Jupiter Saturn Caelus Neptune (i) Find the lower bound for the number of crates required to fit all statues. (ii) Use the first fit decreasing algorithm to estimate the minimum number of crates required to ship all statues. Does the first fit decreasing algorithm give an optimal solution? (iii) Find by trial and error a solution needing only & crates.

could you just do question 4 please