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Branch and Bound Algorithm
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DSA for Beginners Learn more about Branch and Bound in DSA Self Paced Course
What is Branch and Bound Algorithm?
Branch and bound is an algorithm design paradigm which is generally used for solving combinatorial optimization problems. These problems are typically exponential in terms of time complexity and may require exploring all possible permutations in worst case. The Branch and Bound Algorithm technique solves these problems relatively quickly.
Example to show working of Branch and Bound Algorithm
Let us consider the 0/1 Knapsack problem to understand Branch and Bound.
There are many algorithms by which the knapsack problem can be solved:
- DP solution for 0/1 Knapsack
- Backtracking Solution for 0/1 Knapsack .
Let’s see the Branch and Bound Approach to solve the 0/1 Knapsack problem : The Backtracking Solution can be optimized if we know a bound on best possible solution subtree rooted with every node. If the best in subtree is worse than current best, we can simply ignore this node and its subtrees. So we compute bound (best solution) for every node and compare the bound with current best solution before exploring the node.
Example bounds used in below diagram are, A down can give $315, B down can $275, C down can $225, D down can $125 and E down can $30.
- Branch and Bound | Set 1 (Introduction with 0/1 Knapsack)
- Branch and Bound | Set 2 (Implementation of 0/1 Knapsack)
- Branch and Bound | Set 3 (8 puzzle Problem)
- Branch And Bound | Set 4 (Job Assignment Problem)
- Branch and Bound | Set 5 (N Queen Problem)
- Branch And Bound | Set 6 (Traveling Salesman Problem)
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Branch and Bound Algorithm for solving Assignment-probleem
I started doing Branch and Bound Algorithm for assignment problem in C++ and i can't find the right solution. First of all assignment problem example: Assignment problem
Ok so each person can be assigned to one job, and the idea is to assign each job to one of the person so that all the jobs are done in the quickest way.
Here is my code so far:
I know i might be out of track. I didn't use lower bound in the beginning, i'm actually a little confused how this algorithm exactly works. So even step by step walktrough through the algorithm would be helpful.
- I forgot to say that my matrix is 2D vector, and setmatrix() and print() functions are just for filling and printing the matrix – whyunomad Jan 21, 2017 at 14:14
- What is your actual question? Is your code not working or are you just looking for an explanation for the assignment problem? – Michiel uit het Broek May 26, 2017 at 12:49
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- Published: December 1975
A branch and bound algorithm for the generalized assignment problem
- G. Terry Ross 1 &
- Richard M. Soland 2
Mathematical Programming volume 8 , pages 91–103 ( 1975 ) Cite this article
This paper describes what is termed the “generalized assignment problem”. It is a generalization of the ordinary assignment problem of linear programming in which multiple assignments of tasks to agents are limited by some resource available to the agents. A branch and bound algorithm is developed that solves the generalized assignment problem by solving a series of binary knapsack problems to determine the bounds. Computational results are cited for problems with up to 4 000 0–1 variables, and comparisons are made with other algorithms.
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V. Balachandran, “An integer generalized transportation model for optimal job assignment in computer networks”, Working Paper 34-72-3, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, Pa. (November, 1972).
A. Charnes, W.W. Cooper, D. Klingman and R. Niehaus, “Static and dynamic biased quadratic multi-attribute assignment models: solutions and equivalents”, Center for Cybernetic Studies, Research Report CS 115, The University of Texas, Austin, Texas (January, 1973).
R.J. Dakin, “A tree search algorithm for mixed integer programming problems”, Computer Journal 8 (3) (1965) 250–255.
A. DeMaio and C. Roveda, “An all zero–one algorithm for a certain class of transportation problems”, Operations Research 19 (6) (1971) 1406–1418.
A.M. Geoffrion, “An improved implicit enumeration approach for integer programming”, Operations Research 17 (3) (1969) 437–454.
A.M. Geoffrion, “Lagrangean relaxation for integer programming”, Mathematical Programming Study 2 (1974) 82–114.
A.M. Geoffrion and G.W. Graves, “Multicommodity distribution system design by benders decomposition”, Management Science 20 (5) (1974) 822–844.
H. Greenberg and R.L. Hegerich, “A branch search algorithm for the knapsack problem”, Management Science 16 (5) (1970) 327–332.
M.D. Grigoriadis, D.T. Tang and L.S. Woo, “Considerations in the optimal synthesis of some communication networks”, Presented at the 45 th Joint National Meeting of the Operations Research Society of America and The Institute of Management Sciences, Boston, Mass., April 22–24, 1974.
D. Gross and C.E. Pinkus, “Optimal allocation of ships to yards for regular overhauls”, Tech. Memorandum 63095, Institute for Management Science and Engineering, The George Washington University, Washington, D.C. (May, 1972).
G.P. Ingargiola and J.F. Korsh, “Reduction algorithm for zero–one single knapsack problems”, Management Science 20 (4) Part I (1973) 460–463.
D. Klingman and J. Stutz, “Computational testing on an integer generalized network code”, Presented at the 45 th Joint National Meeting of the Operations Research Society of America and The Institute of Management Sciences, Boston, Mass., April 22–24, 1974.
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V. Srinivasan and G. Thompson, “An algorithm for assigning uses to sources in a speical class of transportation problems”, Operations Research 21 (1) (1973) 284–295.
Authors and affiliations.
University of Massachusetts, Amherst, Mass., USA
G. Terry Ross
University of Texas, Austin, Tex., USA
Richard M. Soland
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This research was partly supported by ONR Contracts N00014-67-A-0126-0008 and N00014-67-A-0126-0009 with the Center for Cybernetic Studies, The University of Texas.
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Ross, G.T., Soland, R.M. A branch and bound algorithm for the generalized assignment problem. Mathematical Programming 8 , 91–103 (1975). https://doi.org/10.1007/BF01580430
Received : 18 September 1973
Revised : 15 October 1974
Issue Date : December 1975
DOI : https://doi.org/10.1007/BF01580430
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Branch And Bound | Set 4 (Job Assignment Problem)
- December 22, 2017
- 3 minute read
- No comments
Let there be N workers and N jobs. Any worker can be assigned to perform any job, incurring some cost that may vary depending on the work-job assignment. It is required to perform all jobs by assigning exactly one worker to each job and exactly one job to each agent in such a way that the total cost of the assignment is minimized.
Let us explore all approaches for this problem.
Solution 1: Brute Force We generate n! possible job assignments and for each such assignment, we compute its total cost and return the less expensive assignment. Since the solution is a permutation of the n jobs, its complexity is O(n!).
Solution 2: Hungarian Algorithm The optimal assignment can be found using the Hungarian algorithm. The Hungarian algorithm has worst case run-time complexity of O(n^3).
Solution 3: DFS/BFS on state space tree A state space tree is a N-ary tree with property that any path from root to leaf node holds one of many solutions to given problem. We can perform depth-first search on state space tree and but successive moves can take us away from the goal rather than bringing closer. The search of state space tree follows leftmost path from the root regardless of initial state. An answer node may never be found in this approach. We can also perform a Breadth-first search on state space tree. But no matter what the initial state is, the algorithm attempts the same sequence of moves like DFS.
Solution 4: Finding Optimal Solution using Branch and Bound The selection rule for the next node in BFS and DFS is “blind”. i.e. the selection rule does not give any preference to a node that has a very good chance of getting the search to an answer node quickly. The search for an optimal solution can often be speeded by using an “intelligent” ranking function, also called an approximate cost function to avoid searching in sub-trees that do not contain an optimal solution. It is similar to BFS-like search but with one major optimization. Instead of following FIFO order, we choose a live node with least cost. We may not get optimal solution by following node with least promising cost, but it will provide very good chance of getting the search to an answer node quickly.
There are two approaches to calculate the cost function:
- For each worker, we choose job with minimum cost from list of unassigned jobs (take minimum entry from each row).
- For each job, we choose a worker with lowest cost for that job from list of unassigned workers (take minimum entry from each column).
In this article, the first approach is followed.
Let’s take below example and try to calculate promising cost when Job 2 is assigned to worker A.
Since Job 2 is assigned to worker A (marked in green), cost becomes 2 and Job 2 and worker A becomes unavailable (marked in red).
Now we assign job 3 to worker B as it has minimum cost from list of unassigned jobs. Cost becomes 2 + 3 = 5 and Job 3 and worker B also becomes unavailable.
Finally, job 1 gets assigned to worker C as it has minimum cost among unassigned jobs and job 4 gets assigned to worker C as it is only Job left. Total cost becomes 2 + 3 + 5 + 4 = 14.
Below diagram shows complete search space diagram showing optimal solution path in green.
Below is its C++ implementation.
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Branch and Bound Search with Examples and Implementation in Python
We’ll try to understand one of the heuristic search techniques in this article. The heuristic technique is a criterion for determining which among several alternatives will be the most effective in achieving a particular goal. Branch and bound search is also known as Uniform Cost Search.
What is the branch and bound search algorithm?
Branch and bound is a search algorithm used for combinatory, discrete, and general mathematical optimization problems. It is comparable to backtracking in that it similarly implements a state-space stream to represent the solution to the problem.
However, it is probably more suited to trying to address optimization problems and only minimization problems, not maximization problems. Statistically speaking, a branch and the bound algorithm find the best solution from the entire search space of possibilities for an NP-Hard problem.
How does the branch and bound search work?
In the branch and bound search strategy, a cost function (denoted by g(X)) is generated that, by using a sequence of operators, assigns a cumulative cost to the path from the start node to the current node X. A cheapest price path already discovered is extended at every step of the search space generation process until we reach the goal state.
Branch and bound search is also referred to as a uniform cost search since it expands the least-cost partial path. The actual distance traveled from the beginning to the current node X, for instance, may be represented as g(X) in the traveling salesman problem.
Steps for the algorithm
If g(X) = 1 for all operators, the branch and bound methodology degenerates into a straightforward breadth-first search. Artificial intelligence considers it to be just as detrimental as depth-first and breadth-first. If we add dynamic programming to it, we can make this better by eliminating redundant paths.
We note that the method typically necessitates creating a solution and evaluating its efficacy. Any technique can be used to develop the answer, and heuristics may be used in testing. The following is the basic structure of an algorithm for developing and testing strategies.
Brand and bound search algorithm in action
To understand the concept more clearly, let’s try to implement the 8 puzzle problem using the branch and bound algorithm. The problem description is given below.
A 3 x 3 board with 8 tiles (each tile has a number ranging from 1 to 8) and a single empty space is provided. The goal is to use the vacant space to arrange the numbers on the tiles so that they match the final arrangement. Four neighboring (left, right, above, and below) tiles can be slid into the available area.
To avoid searching in sub-trees that do not include an answer node, the search for an answer node can frequently be sped up using an approximation of the cost function. However, instead of using the backtracking method, it does a BFS-style search.
Basically, Branch and Bound involve three different kinds of nodes.
- A live node is a generated node whose children have not yet been produced.
- The children of the E-node, a live node, are now being examined. Or to put it another way, an E-node is a node that is currently expanding.
- A created node that is not to be developed or examined further is referred to as a dead node. A dead node has already extended all of its children.
Cost function: In the search tree, each node X has a corresponding cost. The next E-node can be found using the cost function. The E-node with the lowest cost is the next one. The definition of the cost function is
Implementing the Branch and Bound Search algorithm in Python
In this article, we have learned one of the most effective algorithms knowns as a branch and bound search. This search algorithm helps to solve many common problems like the N-Queen problem, 0-1 Knapsack Problem, Traveling salesman problem, etc. The algorithm is bit modified in each case according to the conditions provided in the problem but the basic idea of the searching method remains the same as explained earlier.
Branch And Bound Algorithm Analysis For Solving Job Assignment Problems
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The Branch & Bound algorithm is a method commonly used to solve optimization problems. Some examples of problems that the Branch & Bound algorithm can solve are Knapsack Problems, Traveling Salesman Problems, Scheduling Problems and many other optimization problems. Job Assignment Problem is one of the optimization problems of how to do n people doing n jobs where one person does one job. This paper will analyze the Branch and Bound Algorithm in solving Job Assignments Problem.