branch and bound technique for assignment problem

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Introduction to Branch and Bound – Data Structures and Algorithms Tutorial

Branch and bound algorithms are used to find the optimal solution for combinatory, discrete, and general mathematical optimization problems. A branch and bound algorithm provide an optimal solution to an NP-Hard problem by exploring the entire search space. Through the exploration of the entire search space, a branch and bound algorithm identify possible candidates for solutions step-by-step.

There are many optimization problems in computer science, many of which have a finite number of the feasible shortest path in a graph or minimum spanning tree that can be solved in polynomial time. Typically, these problems require a worst-case scenario of all possible permutations. The branch and bound algorithm create branches and bounds for the best solution.

In this tutorial, we’ll discuss the branch and bound method in detail.

Different search techniques in branch and bound:

The Branch  algorithms incorporate different search techniques to traverse a state space tree. Different search techniques used in B&B are listed below:

1. LC search (Least Cost Search):

It uses a heuristic cost function to compute the bound values at each node. Nodes are added to the list of live nodes as soon as they get generated. The node with the least value of a cost function selected as a next E-node.

2.BFS(Breadth First Search): It is also known as a FIFO search. It maintains the list of live nodes in first-in-first-out order i.e, in a queue, The live nodes are searched in the FIFO order to make them next E-nodes.

3. DFS (Depth First Search): It is also known as a LIFO search. It maintains the list of live nodes in last-in-first-out order i.e. in a stack.

The live nodes are searched in the LIFO order to make them next E-nodes.

Introduction to Branch and Bound

When to apply Branch and Bound Algorithm?

Branch and bound is an effective solution to some problems, which we have already discussed. We’ll discuss all such cases where branching and binding are appropriate in this section.

• It is appropriate to use a branch and bound approach if the given problem is discrete optimization. Discrete optimization refers to problems in which the variables belong to the discrete set. Examples of such problems include 0-1 Integer Programming and Network Flow problems.
• When it comes to combinatory optimization problems, branch and bound work well. An optimization problem is optimized by combinatory optimization by finding its maximum or minimum based on its objective function. The combinatory optimization problems include Boolean Satisfiability and Integer Linear Programming.

Basic Concepts of Branch and Bound:

▸ Generation of a state space tree:

As in the case of backtracking, B&B generates a state space tree to efficiently search the solution space of a given problem instance.

In B&B, all children of an E-node in a state space tree are produced before any live node gets converted in an E-node. Thus, the E-node remains an E-node until i becomes a dead node.

• Evaluation of a candidate solution:

Unlike backtracking, B&B needs additional factors evaluate a candidate solution:

• A way to assign a bound on the best values of the given criterion functions to each node in a state space tree: It is produced by the addition of further components to the partial solution given by that node.
• The best values of a given criterion function obtained so far: It describes the upper bound for the maximization problem and the lower bound for the minimization problem.
• A feasible solution is defined by the problem states that satisfy all the given constraints.
• An optimal solution is a feasible solution, which produces the best value of a given objective function.
• Bounding function :

It optimizes the search for a solution vector in the solution space of a given problem instance.

It is a heuristic function that evaluates the lower and upper bounds on the possible solutions at each node. The bound values are used to search the partial solutions leading to an optimal solution. If a node does not produce a solution better than the best solution obtained thus far, then it is abandoned without further exploration.

The algorithm then branches to another path to get a better solution. The desired solution to the problem is the value of the best solution produced so far.

▸ The reasons to dismiss a search path at the current node :

(i) The bound value of the node is lower than the upper bound in the case of the maximization problem and higher than the lower bound in the case of the minimization problem. (i.e. the bound value of the ade is not better than the value of the best solution obtained until that node).

(ii) The node represents infeasible solutions, de violation of the constraints of the problem.

(iii) The node represents a subset of a feasible solution containing a single point. In this case, if the latest solution is better than the best solution obtained so far the best solution is modified to the value of a feasible solution at that node.

Types of Branch and Bound Solutions:

The solution of the Branch and the bound problem can be represented in two ways:

• Variable size solution: Using this solution, we can find the subset of the given set that gives the optimized solution to the given problem. For example, if we have to select a combination of elements from {A, B, C, D} that optimizes the given problem, and it is found that A and B together give the best solution, then the solution will be {A, B}.
• Fixed-size solution: There are 0s and 1s in this solution, with the digit at the ith position indicating whether the ith element should be included, for the above example, the solution will be given by {1, 1, 0, 0}, here 1 represent that we have select the element which at ith position and 0 represent we don’t select the element at ith position.

Classification of Branch and Bound Problems:

The Branch and Bound method can be classified into three types based on the order in which the state space tree is searched. 

• FIFO Branch and Bound
• LIFO Branch and Bound
• Least Cost-Branch and Bound

We will now discuss each of these methods in more detail. To denote the solutions in these methods, we will use the variable solution method.

1. FIFO Branch and Bound

First-In-First-Out is an approach to the branch and bound problem that uses the queue approach to create a state-space tree. In this case, the breadth-first search is performed, that is, the elements at a certain level are all searched, and then the elements at the next level are searched, starting with the first child of the first node at the previous level.

For a given set {A, B, C, D}, the state space tree will be constructed as follows :

State Space tree for set {A, B, C, D}

The above diagram shows that we first consider element A, then element B, then element C and finally we’ll consider the last element which is D. We are performing BFS while exploring the nodes.

So, once the first level is completed. We’ll consider the first element, then we can consider either B, C, or D. If we follow the route then it says that we are doing elements A and D so we will not consider elements B and C. If we select the elements A and D only, then it says that we are selecting elements A and D and we are not considering elements B and C.

Selecting element A

Now, we will expand node 3, as we have considered element B and not considered element A, so, we have two options to explore that is elements C and D. Let’s create nodes 9 and 10 for elements C and D respectively.

Considered element B and not considered element A

Now, we will expand node 4 as we have only considered elements C and not considered elements A and B, so, we have only one option to explore which is element  D. Let’s create node 11 for D.

 Considered elements C and not considered elements A and B

Till node 5, we have only considered elements D, and not selected elements A, B, and C. So, We have no more elements to explore, Therefore on node 5, there won’t be any expansion.

Now, we will expand node 6 as we have considered elements A and B, so, we have only two option to explore that is element C and D. Let’s create node 12 and 13 for C and D respectively.

Expand node 6

Now, we will expand node 7 as we have considered elements A and C and not consider element B, so, we have only one option to explore which is element  D. Let’s create node 14 for D.

Expand node 7

Till node 8, we have considered elements A and D, and not selected elements B and C, So, We have no more elements to explore, Therefore on node 8, there won’t be any expansion.

Now, we will expand node 9 as we have considered elements B and C and not considered element A, so, we have only one option to explore which is element  D. Let’s create node 15 for D.

Expand node 9

2. LIFO Branch and Bound

The Last-In-First-Out approach for this problem uses stack in creating the state space tree. When nodes are added to a state space tree, they are added to a stack. After all nodes of a level have been added, we pop the topmost element from the stack and explore it.

State space tree for element {A, B, C, D}

Now the expansion would be based on the node that appears on the top of the stack. Since node 5 appears on the top of the stack, so we will expand node 5. We will pop out node 5 from the stack. Since node 5 is in the last element, i.e., D so there is no further scope for expansion.

The next node that appears on the top of the stack is node 4. Pop-out node 4 and expand. On expansion, element D will be considered and node 6 will be added to the stack shown below:

Expand node 4

The next node is 6 which is to be expanded. Pop-out node 6 and expand. Since node 6 is in the last element, i.e., D so there is no further scope for expansion.

The next node to be expanded is node 3. Since node 3 works on element B so node 3 will be expanded to two nodes, i.e., 7 and 8 working on elements C and D respectively. Nodes 7 and 8 will be pushed into the stack.

The next node that appears on the top of the stack is node 8. Pop-out node 8 and expand. Since node 8 works on element D so there is no further scope for the expansion.

Expand node 3

The next node that appears on the top of the stack is node 7. Pop-out node 7 and expand. Since node 7 works on element C so node 7 will be further expanded to node 9 which works on element D and node 9 will be pushed into the stack.

The next node that appears on the top of the stack is node 9. Since node 9 works on element D, there is no further scope for expansion.

The next node that appears on the top of the stack is node 2. Since node 2 works on the element A so it means that node 2 can be further expanded. It can be expanded up to three nodes named 10, 11, 12 working on elements B, C, and D respectively. There new nodes will be pushed into the stack shown as below:

Expand node 2

In the above method, we explored all the nodes using the stack that follows the LIFO principle.

3. Least Cost-Branch and Bound

To explore the state space tree, this method uses the cost function. The previous two methods also calculate the cost function at each node but the cost is not been used for further exploration.

In this technique, nodes are explored based on their costs, the cost of the node can be defined using the problem and with the help of the given problem, we can define the cost function. Once the cost function is defined, we can define the cost of the node. Now, Consider a node whose cost has been determined. If this value is greater than U0, this node or its children will not be able to give a solution. As a result, we can kill this node and not explore its further branches. As a result, this method prevents us from exploring cases that are not worth it, which makes it more efficient for us. 

Let’s first consider node 1 having cost infinity shown below:

In the following diagram, node 1 is expanded into four nodes named 2, 3, 4, and 5.

Node 1 is expanded into four nodes named 2, 3, 4, and 5

Assume that cost of the nodes 2, 3, 4, and 5 are 12, 16, 10, and 315 respectively. In this method, we will explore the node which is having the least cost. In the above figure, we can observe that the node with a minimum cost is node 4. So, we will explore node 4 having a cost of 10.

During exploring node 4 which is element C, we can notice that there is only one possible element that remains unexplored which is D (i.e, we already decided not to select elements A, and B). So, it will get expanded to one single element D, let’s say this node number is 6.

Exploring node 4 which is element C

Now, Node 6 has no element left to explore. So, there is no further scope for expansion. Hence the element {C, D} is the optimal way to choose for the least cost.

Problems that can be solved using Branch and Bound Algorithm:

The Branch and Bound method can be used for solving most combinatorial problems. Some of these problems are given below:

Advantages of Branch and Bound Algorithm:

• We don’t explore all the nodes in a branch and bound algorithm. Due to this, the branch and the bound algorithm have a lower time complexity than other algorithms.
• Whenever the problem is small and the branching can be completed in a reasonable amount of time, the algorithm finds an optimal solution.
• By using the branch and bound algorithm, the optimal solution is reached in a minimal amount of time. When exploring a tree, it does not repeat nodes.

Disadvantages of Branch and Bound Algorithm:

• It takes a long time to run the branch and bound algorithm. 
• In the worst-case scenario, the number of nodes in the tree may be too large based on the size of the problem.

The branch and bound algorithms are one of the most popular algorithms used in optimization problems that we have discussed in our tutorial. We have also explained when a branch and bound algorithm is appropriate for a user to use. In addition, we presented an algorithm based on branches and bounds for assigning jobs. Lastly, we discussed some advantages and disadvantages of branch and bound algorithms.

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Branch and Bound

I. introduction, ii. illustration on the job assignment problem, iii. the general branch and bound algorithm, iv. criteria for the choice of approximate cost functions, v. implementation of the b&b job assignment algorithm, the general branch and bound algorithm.

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Branch and Bound Technique

Algorithms branch and bound.

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With this article, you have explained the idea of Branch and Bound Technique, types of Branch and Bound Technique and applications of the technique.

Introduction to Branch and Bound Technique

The algorithm: branch and bound technique.

• Types of Solution

Types of Branch and Bound

Why use branch and bound, problems that can be solved using branch and bound.

The Branch and Bound Technique is a problem solving strategy, which is most commonly used in optimization problems, where the goal is to minimize a certain value. The optimized solution is obtained by means of a state space tree (A state space tree is a tree where the solution is constructed by adding elements one by one, starting from the root. Note that the root contains no element). This method is best when used for combinatorial problems with exponential time complexity, since it provides a more efficient solution to such problems.

In this technique, the first step is to create a function U (which represents an upper bound to the value that node and its children shall achieve), that we intend to minimize. We call this function the objective function. Note that the branch and bound technique can also be used for maximization problems, since multiplying the objective function by -1 coverts the problem to a minimization problem. Let this function have an initial maximum value, according to the conditions of the given problem. Also, let U 0 be the initial value of U . We also calculate a cost function C which will give us the exact value that the particular node shall achieve.

The next question is the order in which the tree is to be searched. For this, there exist multiple types of branch and bound, which we shall discuss in detail later. For now, let us assume that there is a set S consisting of subsets of the given set of values in the order in which they need to be searched.

The algorithm of the branch and bound method for this problem will be as follows:

For each subset s in S, do the following:

• Calculate the value of U(s) .
• If U(s) U 0 , then U 0 = U(s).

In the end, the the subset s for which the current value of U 0 is obtained will be the best solution to the given problem, and the value of the cost function at that node will give us the solution. Here, after each level, the value of U 0 tells us that there shall be a node with cost less than that value.

Types of Solutions

For a branch and bound problem, there are two ways in which the solution can be represented:

Variable size solution : This solution provides the subset of the given set that gives the optimized solution for the given problem. For example, if the we are to select a combination of elements from {A, B, C, D, E} that optimizes the given problem, and it is found that A, B, and E together give the best solution, then the solution will be {A, B, E}.

Fixed size solution : This solution is a sequence of 0s and 1s, with the digit at the i th position denoting whether the i th element should be included or not. Hence, for the earlier example, the solution will be given by {1, 1, 0, 0, 1}.

There are multiple types of the Branch and Bound method, based on the order in which the state space tree is to be searched. We shall now discuss each of these methods in detail. We will be using the variable solution method to denote the solutions in these methods.

FIFO Branch and Bound

The First-In-First-Out approach to the branch and bound problem follows the queue approach in creating the state-space tree. Here, breadth first search is performed, i.e., the elements at a particular level are all searched, and then the elements of the next level are searched, starting from the first child of the first node in the previous level.

For a given set {A, B, C, D}, the state space tree will be constructed as follows :

Here, note that the number assigned to the node signifies the order in which the tree shall be constructed. The element next to the set denotes the next element to be added to the subset. Note that if an element is getting added, it is assumed here that all elements in the set preceeding that element are not added. For example, in node 4, D is getting added. This implies that elements A, B and C are not added.

LIFO Branch and Bound

The Last-In-First-Out approach to this problem follows the stack approach in creating the state space tree. Here, when nodes get added to the state space tree, think of them as getting added to a stack. When all nodes of a level are added, we pop the topmost element from the stack and then explore it. Hence, the state space tree for the same example {A, B, C, D} will be as follows:

Here, one can see that the main difference lies in the order in which the nodes have been explored.

Least Cost-Branch and Bound

This method of exploration uses the cost function in order to explore the state space tree. Although the previous two methods calculate the cost function at each node, this is not used as a criterion for further exploration.

In this method, after the children of a particular node have been explored, the next node to be explored would be that node out of the unexplored nodes which has the least cost. For example, in the previous example, after reaching node 5, the next node to be explored would be that which has the least cost among nodes 2, 3, 4, 5.

The Branch and Bound method is preferred over other similar methods such as backtracking when it comes to optimization problems. Here, the cost and the objective function help in finding branches that need not be explored.

Suppose the cost of a particular node has been determined. If this value is greater than that of U 0 , this means that there is no way this node or its children shall give a solution. Hence, we can kill this node and not explore its further branches. This method helps us rule out cases not worth exploring, and is therefore more efficient.

The Branch and Bound method can be used for solving most combinatorial problems. Some of these problems are given below:

Job Sequencing : Suppose there is a set of N jobs and a set of N workers. Each worker takes a specific time to complete each of the N jobs. The job sequencing problem deals with finding that order of the workers, which minimizes the time taken to complete the job.

0/1 Knapsack problem : Given a set of weights of N objects and a sack which can carry a maximum of W units. The problem deals with finding that subset of items such that the maximum possible weight is carried in the sack. Here, one cannot take part of an object, i.e., one can either take an object or not take it. This problem can also be solved using the backtracking, brute force and the dynamic programming approach.

Traveling Salesman Problem : Here, we are given a set of N cities, and the cost of traveling between all pairs of cities. The problem is to find a path such that one starts from a given node, visits all cities exactly once, and returns back to the starting city.

With this article at OpenGenus, you must have the complete idea of Branch and Bound Technique.

A branch and bound algorithm for the generalized assignment problem

• Published: December 1975
• Volume 8 , pages 91–103, ( 1975 )

Cite this article

• G. Terry Ross 1 &
• Richard M. Soland 2  

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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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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

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Issue Date : December 1975

DOI : https://doi.org/10.1007/BF01580430

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Branch and Bound Algorithm

Introduction

Branch and bound is a method of designing  algorithms commonly used to solve combinatorial optimization problems . In the worst-case scenario, these problems are typically exponential in terms of  time complexity and may necessitate exploring all possible permutations. These problems are solved relatively quickly using the Branch and Bound Algorithm . It is a process that involves systematically enumerating candidate solutions through a state space search. The set of candidate solutions is considered as a tree with the full set at the root. The algorithm explores branches of this tree that represent subsets of the solution set.

What is Branch and Bound Algorithm?

One of the problem-solving techniques is a branch and bound. It is similar to backtracking in that it employs the state space tree. It is engaged in the solution of optimisation and minimisation problems. If given a maximisation problem, we can use the Branch and bound technique to convert it by simply altering the issue to a maximisation problem.

Branch and bound algorithms solve problems with combinatorial, discrete, and general mathematical optimisation. Given an NP-Hard situation, a branch and bound algorithm explores the entire search space of possible solutions and provides an optimal solution.

A branch and bound algorithm enumerates possible candidate solutions step by step by exploring the whole search space. We begin by creating a rooted decision tree with all possible solutions. The root node represents the entire search space:

Each child node, in this case, is a component of the solution set and a partial solution. Before creating the rooted decision tree, we created an upper and lower bound for a given problem based on the optimal solution. We must choose which nodes to include in the solution set at each level. We look into the node with the highest quality bound for each group. This allows us to identify the most suitable solution quickly.

Finding an excellent upper and lower bound in these situations is crucial. An upper bound can be discovered by using any local optimisation technique or location in the search space.

On the other hand, convex relaxation or duality can offer a lower bound.

The solution set should generally be split up into smaller subsets.

We first construct a rooted decision tree to find the best solution set and then choose the best subset (node) at each level.

Branch and Bound Algorithm Example

In the below section, we are going to discuss how the job assignment problem can be solved using a branch and bound algorithm.

Statement of the Problem

Let us begin by defining a job assignment problem. There can be N jobs and N workers in a standard version of a job assignment problem. To keep things simple, we'll use three jobs and three workers in our example:

We can assign any available job to any worker, with the caveat that if a job is posted to a worker, the other workers are not allowed to take that job. We should also note that each job has a cost, which varies from worker to worker.

The main goal here is to complete all jobs by assigning one job to each worker to minimise the total cost of all jobs.

Pseudocode for the Branch and Bound Algorithm

Let's look at how to use a branch and bound algorithm to solve the job assignment problem.

Let's start with the pseudocode:

The input cost matrix in this instance is called MinCost, and it includes data like the number of open jobs, a list of open workers, and the cost for each job. The active nodes are tracked by the function MinCost(). The minimum cost of the active node at each level of the tree is determined by the function LeastCost(). We return the lowest-cost node after finding it and removing it from the active nodes list.

In the pseudocode, we're using the Add() function, which calculates the cost of a specific node and adds it to the list of active nodes.

Each node in the search space tree contains some information, such as cost, the total number of jobs, and the total number of workers.

We initially had mathsf3 jobs available. Employee A has the option of taking any of the available jobs. So, at the mathsf1 level, we assigned all available jobs to worker A and calculated the cost. We can see that assigning jobs mathsf2 to worker A results in the lowest cost in level mathsf1 of the search space tree. So we assign the job  A and proceed with the algorithm. "Yes" indi-prices that this is the current best price.

We still hagive open jobs after assigning the job mathsf2 to worker A. Consider worker B for a moment. To save money, we're attempting to post either job mathsf1 or job mathsf3 to worker B.

We can either assign the job mathsf1 or mathsf3 to worker B. We recheck the cost and give job mathbfmathsf1 to worker B because it is the cheapest in level mathbfmathsf2.

Finally, we trust worker C with the job mathbfmathsf3, and the optimal cost is 12.

Frequently Asked Questions

Which technique solves problems faster in branch and bound.

The Best First Search (BFS) technique, used in the Branch and Bound method, often solves problems faster. It prioritizes branches with promising cost estimates, leading to an expedited optimal solution discovery.

Can a maximization problem can be solved by branch and bound technique?

Yes, the Branch and Bound technique can solve maximization problems. By establishing upper and lower bounds for the optimal solution, the technique efficiently prunes the search space, helping identify the maximum value.

How branch and bound is better than backtracking?

Branch and Bound is often better than backtracking because it can discard many subproblems without explicitly solving them, due to its bounding function. This pruning of the search space typically makes it more efficient than backtracking.

The branch and bound algorithm are one of the most commonly used  algorithms optimisation problems. This topic has been thoroughly covered in this tutorial.

We've discussed when a branch and bound algorithm is the best option for a user. In addition, we presented a branch and bound algorithm for resolving the job assignment problem.

Recommended Readings:

• Kadane's Algorithm
• Greedy Layer-wise Pre-Training
• Unbounded Knapsack
• Backtracking Algorithm
• Design and Analysis of Algorithms

Finally, we discussed the branch and bound algorithm's benefits and drawbacks.

Do check out our blogs on  object-oriented programming and  Data Structure . 

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Optimization methods in management science, tutorial 10: branch and bound.

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Asymptotic analysis, analysis of sorting, divide and conquer, lower bound theory, sorting in linear time, binary search trees, red black tree, dynamic programming, greedy algorithm, backtracking, shortest path, all-pairs shortest paths, maximum flow, sorting networks, complexity theory, approximation algo, string matching.

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