Answered: The goal of the maximal flow problem is…

The goal of the maximal flow problem is to maximize the amount of flow of items from an origin to a destination. Discuss

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter8: Network Models

Section8.3: Maximum-flow Problems

Problem 12P

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Use the worksheets to show, one path augmentation at a time, how to use the Ford-Fulkerson Algorithm to compute a max flow and a min cut for the flow network. As you go along, write the flows for each edge in the little squares. When you reach the end of the algorithm, shade in the nodes that are on the "A side" of the minimum cut. You may want to review the Ford-Fulkerson Algorithm before starting on the problem.
In the diagram attached, a flow network has been depicted. Perform the following tasks using the Push Re-label Algorithm: - Optimize the maximum flow between S1 to t2 S2 to t1 S1 & S2 to t2 S1 & S2 to t1 Evaluate your final results for any shortcomings, and propose changes to solve them. 12 14 34 33 16 18 30 Note: Please assign the values (individual digits - one digit per edge) from your mobile number (0967910053) to the unmarked edges. 10 unmarked edges will be given with 10 digits in your mobile number and any 'O digit should be replaced with value '10'
Suppose that there are m students who want to take part in n projects. A student is allowed to join in a particular project only if the student is qualified for the project. Each project can only have a limited number of students, and each student can take part in at most one project. The goal is to maximize the number of students that are admitted to the projects. Show how to solve this problem by transforming it into a maximum flow problem. What is the running time of your algorithm?
2. Consider the following problem: You have three jugs with capacities 4, 5 and 6 litters respectively. Initially all of the jugs contain 2 litters of water. At any time-step, you can either remove 1 litter of water from any 2 jugs or add 1 litter to any 1 jug. You have to continue the process until the jugs contain 8 or more litters of water in total. The target is to reach the goal within minimum number of steps. Answer the following: a. How many variables are required to mathematically represent the states of the problem? b. What is the size of the state-space? Explain your calculation. c. List any 30 goal states. d. Will a DFS find optimal solution for this problem? What about a BFS? Which one would you go for, and why?
We want to send n cargo by n truck. Each truck can move only one cargo and receives different costs for moving each of these cargoes. We want to move cargo with trucks so that we pay the lowest cost. First, write a backtracking algorithm for this purpose and then, explain How to reduce the branch and the complexity of the algorithm with an algorithm(Pseudocode is required for backtracking algorithm. In the branch and limit algorithm, it is sufficient to explain the limit function and the method of pruning.)
Write the algorithm that finds and returns how many paths in k units of length between any given two nodes (source node, destination node; source and target nodes can also be the same) in a non-directional and unweighted line of N nodes represented as a neighborhood matrix. (Assume that each side in the unweighted diagram is one unit long.) Note: By using the problem reduction method of the Transform and Conquer strategy, you have to make the given problem into another problem. Algorithm howManyPath (M [0..N-1] [0..N-1], source, target, k)// Input: NxN neighborhood matrix, source, target nodes, k value.// Ouput: In the given line, there are how many different paths of k units length between the given source and target node.
Problem 1.2 Use solve_ivp to integrate the equations of motion of the two-body problem. The initial position and velocity state vector Xo = (r, v)™ is: and 3]: # YOUR CODE HERE r = raise NotImplementedError() V = The duration of the simulation is for one day but you should compute the solutuion for every ten seconds. We show you how to control this below but you will have to make changes to this code to get it to work correctly for this simiualtion. Not ImplementedError Cell In [3], line 2 Lastly, when using solve_ivp make sure that you use rtol=1e-12 and atol [6510.759869015321 2676.16546382759 km 333.334029373109 [-2.232024608624287 9.49860960555864 km/s 1.18311436869621 1 # YOUR CODE HERE ----> 2 raise NotImplementedError() = 1e-12. Traceback (most recent call last)
A Consider the minimum flow problem shown below, where the source is node A, the sink is node F, and the arc capacities are the numbers shown next to these directed arcs. 7 B C 2 4 7 6 D E 6 F Iteration Augmenting path 1 2 3 Use the Augmenting Path Algorithm (Ford-Fulkerson Method) for the Minimum Flow Problem to find the flow pattern giving the minimum flow from the source to the sink. Formulate this problem as a minimum cost flow problem by showing the appropriate network representation, including adding the arc A → F. Use F = 16. List the augmenting path and c* for each iteration in the above table. Show your final result by either listing the optimal flow assignment paths or clearly labeling the flow on the network. C*
F Source For the network shown below, the arc capacity from node i to node j is the number nearest node i along the arc between these nodes. Use the augmenting path algorithm described in Sec. 10.5 to find the flow pattern giving the maximum flow from the source to the sink. List the augmenting path and c* for each iteration in a table given below. Show your result by either listing the optimal flow assignment paths or clearly labeling the flow on the network. 1 4 24 3 33 4 4 54 6 7 Sink F Iteration Augmenting path 1 2 3 C*
A robot can move horizontally or vertically to any square in the same row or in the same column of a board. Find the number of the shortest paths by which a robot can move from one corner of a board to the diagonally opposite corner. The length of a path is measured by the number of squares it passes through, including the ﬁrst and the least squares. Write the recurrence relation if you solve the problem by a dynamic programming algorithm.
Please answer the following question in detail and explain all the proofs and assumptions for all parts. The question has three parts, (a), (b) and (c). Iterative lengthening search is an iterative analogue of uniform-cost search. The basic idea is to use increasing limits on path cost. If a node is generated whose path cost exceeds the current limit, it is immediately discarded. For each new iteration, the limit is set to the lowest path cost of any node discarded in the previous iteration. (a) Show that this algorithm is optimal for general path costs. You may assume that all costs are integers (this is not a loss of generality if the search space is finite). You may wish to consider the minimal path cost C; what happens when we set the path cost to be some limit l < C? (b) Consider a uniform tree with branching factor b, solution depth d, and unit step costs (each action costs one unit). How many iterations will iterative lengthening require? (c) (7 points) Now consider the…