arc. Consider the network of Figure 2, where the capacities of arcs are given in rectangles at each (i) Knowing that (W, W) with W = network. {s, a, b, c} is a minimal s- t cut suggest a maximal flow for this 100 80 60 50 70 30 50 b 40 d 80 100 Figure 2: Network to be considered in Combinatorial Optimisation Problem 3 Hints: The flow can be suggested by copying the network of Figure 2 and indicating the flow through each arc as done on Figure 1. The flow can be found by using the relations between maximal flows and minimal cuts and without applying the Ford-Fulkerson algorithm. (ii) Justify that the suggested flow is maximal by running one iteration of the Ford-Fulkerson algorithm ending with nonbreakthrough. Show your working. (iii) Suppose that f is a maximal flow through the network. Find all arcs e for which the value f(e) is the same for all maximal flows, and all arcs e for which it can be different in a different maximal flow. Justify. (iv) Suppose that we are required to increase the value of the maximal flow by 30, and that we need to achieve this by increasing the capacity of just one arc. Which arc (or arcs) can be used for this task? For any such arc that you find, draw the resulting network and indicate the maximal flow. For the remaining arcs, give your argument without finding any maximal flows why these arcs cannot be used for this task.

Consider noncoplanar points A, B, C, and D. Using three points at a time such as A, B, and C, how many planes are determined by these points?

use the map in Figure 11.2. a Find the grid section of Taylor Hall. b. What is located in section 3B?

Suppose that Mrs. Jena (in the position of T) want to travel to the top of regions diagonally right or left. It is impossible to stop on the STOP region and also not possible to jump too. Determine the possible number of routes that she is travelling to the top region. STOP T

Please do the folloowing questions with handwritten working out

Question 3 Consider the directed network (D, c) given by the following drawing, where each arc e Є A(D) is labelled by its capacity c(e) and two vertices s and t have been identified. S ~ 2 19 5 ~ 1 3 00 4 3 w (b) Give a minimum s-t-cut of (D, c). Justify your answer and give the capacity of the cut.