job assignment problem in c

Branch and Bound Algorithm

Last updated: March 18, 2024

1. Overview

In computer science, there is a large number of optimization problems which has a finite but extensive number of feasible solutions. Among these, some problems like finding the shortest path in a graph  or  Minimum Spanning Tree  can be solved in polynomial time .

A significant number of optimization problems like production planning , crew scheduling can’t be solved in polynomial time, and they belong to the NP-Hard class . These problems are the example of NP-Hard combinatorial optimization problem .

Branch and bound (B&B) is an algorithm paradigm widely used for solving such problems.

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

2. Basic Idea

Branch and bound algorithms are used to find the optimal solution for combinatory, discrete, and general mathematical optimization problems. In general, given an NP-Hard problem, a branch and bound algorithm explores the entire search space of possible solutions and provides an optimal solution.

A branch and bound algorithm consist of stepwise enumeration of possible candidate solutions by exploring the entire search space. With all the possible solutions, we first build a rooted decision tree. The root node represents the entire search space:

Here, each child node is a partial solution and part of the solution set. Before constructing the rooted decision tree, we set an upper and lower bound for a given problem based on the optimal solution. At each level, we need to make a decision about which node to include in the solution set. At each level, we explore the node with the best bound. In this way, we can find the best and optimal solution fast.

Now it is crucial to find a good upper and lower bound in such cases. We can find an upper bound by using any local optimization method or by picking any point in the search space. On the other hand, we can obtain a lower bound from convex relaxation  or  duality .

In general, we want to partition the solution set into smaller subsets of solution. Then we construct a rooted decision tree, and finally, we choose the best possible subset (node) at each level to find the best possible solution set.

3. When Branch and Bound Is a Good Choice?

We already mentioned some problems where a branch and bound can be an efficient choice over the other algorithms. In this section, we’ll list all such cases where a branch and bound algorithm is a good choice.

If the given problem is a discrete optimization problem, a branch and bound is a good choice. Discrete optimization is a subsection of optimization where the variables in the problem should belong to the discrete set. Examples of such problems are 0-1 Integer Programming  or  Network Flow problem .

Branch and bound work efficiently on the combinatory optimization problems. Given an objective function for an optimization problem, combinatory optimization is a process to find the maxima or minima for the objective function. The domain of the objective function should be discrete and large. Boolean Satisfiability , Integer Linear Programming are examples of the combinatory optimization problems.

4. Branch and Bound Algorithm Example

In this section, we’ll discuss how the job assignment problem can be solved using a branch and bound algorithm.

4.1. Problem Statement

	Job 1	Job 2	Job 3
A	9	3	4
B	7	8	4
C	10	5	2

We can assign any of the available jobs to any worker with the condition that if a job is assigned to a worker, the other workers can’t take that particular job. We should also notice that each job has some cost associated with it, and it differs from one worker to another.

Here the main aim is to complete all the jobs by assigning one job to each worker in such a way that the sum of the cost of all the jobs should be minimized.

4.2. Branch and Bound Algorithm Pseudocode

Now let’s discuss how to solve the job assignment problem using a branch and bound algorithm.

Let’s see the pseudocode first:

In the search space tree, each node contains some information, such as cost, a total number of jobs, as well as a total number of workers.

Now let’s run the algorithm on the sample example we’ve created:

4. Advantages

In a branch and bound algorithm, we don’t explore all the nodes in the tree. That’s why the time complexity of the branch and bound algorithm is less when compared with other algorithms.

If the problem is not large and if we can do the branching in a reasonable amount of time, it finds an optimal solution for a given problem.

The branch and bound algorithm find a minimal path to reach the optimal solution for a given problem. It doesn’t repeat nodes while exploring the tree.

5. Disadvantages

The branch and bound algorithm are time-consuming. Depending on the size of the given problem, the number of nodes in the tree can be too large in the worst case.

Also, parallelization is extremely difficult in the branch and bound algorithm.

6. Conclusion

One of the most popular algorithms used in the optimization problem is the branch and bound algorithm. We’ve discussed it thoroughly in this tutorial.

We’ve explained when a branch and bound algorithm would be the right choice for a user to use. Furthermore, we’ve presented a branch and bound based algorithm for solving the job assignment problem.

Finally, we mentioned some advantages and disadvantages of the branch and bound algorithm.

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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.

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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.

Complete Algorithm:

Below is its C++ implementation.

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Venkatesan Prabu

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a generalized job assignment problem

The following problem is from a past algorithms course exam and I'm using it to test my knowledge.

There are m machines and n jobs. Each machine can doing a subset of jobs. Each machine i has a capacity $C_i$ , meaning that it has $C_i$ units of processing time. Each job $j$ has a demand $D_j$ , meaning that it requires $D_j$ units of processing time to complete. We'd like to assign all the jobs to the machines, so that each job is assigned to only one machine, and no machine is overloaded (i.e. the total demands assigned to machine i doesn't exceed its capacity $C_i$ ).

Input: $m$ positive numbers $C_1,\cdots, C_m$ , n positive numbers $D_1,\cdots, D_n$ , and for each $1\leq i\leq m$ and $1\leq j\leq n,$ a boolean variable $x_{i,j}$ indicating whether machine $i$ can do job $j$ .

Output: Does there exist an assignment such that all the jobs are assigned to machines, so that each job is assigned to only one machine and no machine is overloaded?

Question: prove the above problem is NP-complete, or give an algorithm to solve the decision problem in polynomial time.

I think the problem might be NP-complete. The decision problem asks whether an assignment assigns at least k jobs, where k is a parameter to the decision problem. Clearly the problem is in NP; one can verify in polynomial time that an assignment satisfies that no machine is overloaded and each job is assigned to one machine. One can do this by checking the jobs assigned to machine i and verifying that the total sum of the $D_j$ 's associated with machine i is at most $C_i$ . One can then check at the same time that no job is assigned to two different machines.

But I'm not sure which NP-complete problem to reduce from. For instance, I know the following well-known problems are NP-complete: vertex cover, 3-SAT, hamiltonian cycle, set cover, hamiltonian path, clique, independent set, 3 coloring, subset sum etc.

Maybe Vertex cover would be useful?

• np-complete

This problem is a generalization of the decision version of the bin packing problem (BPP) . While all bins in BPP have the same given capacity, the capacities of the machines in this problem are variable.

The decision version of BPP is $\mathsf{NP}$ -complete. So you have guessed correctly: this problem is $\mathsf{NP}$ -complete since this problem is in $\mathsf{NP}$ as proved in the question.

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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.

Solving assignment problem using min-cost-flow ¶

The assignment problem has two equivalent statements:

• Given a square matrix $A[1..N, 1..N]$ , you need to select $N$ elements in it so that exactly one element is selected in each row and column, and the sum of the values of these elements is the smallest.
• There are $N$ orders and $N$ machines. The cost of manufacturing on each machine is known for each order. Only one order can be performed on each machine. It is required to assign all orders to the machines so that the total cost is minimized.

Here we will consider the solution of the problem based on the algorithm for finding the minimum cost flow (min-cost-flow) , solving the assignment problem in $\mathcal{O}(N^3)$ .

Description ¶

Let's build a bipartite network: there is a source $S$ , a drain $T$ , in the first part there are $N$ vertices (corresponding to rows of the matrix, or orders), in the second there are also $N$ vertices (corresponding to the columns of the matrix, or machines). Between each vertex $i$ of the first set and each vertex $j$ of the second set, we draw an edge with bandwidth 1 and cost $A_{ij}$ . From the source $S$ we draw edges to all vertices $i$ of the first set with bandwidth 1 and cost 0. We draw an edge with bandwidth 1 and cost 0 from each vertex of the second set $j$ to the drain $T$ .

We find in the resulting network the maximum flow of the minimum cost. Obviously, the value of the flow will be $N$ . Further, for each vertex $i$ of the first segment there is exactly one vertex $j$ of the second segment, such that the flow $F_{ij}$ = 1. Finally, this is a one-to-one correspondence between the vertices of the first segment and the vertices of the second part, which is the solution to the problem (since the found flow has a minimal cost, then the sum of the costs of the selected edges will be the lowest possible, which is the optimality criterion).

The complexity of this solution of the assignment problem depends on the algorithm by which the search for the maximum flow of the minimum cost is performed. The complexity will be $\mathcal{O}(N^3)$ using Dijkstra or $\mathcal{O}(N^4)$ using Bellman-Ford . This is due to the fact that the flow is of size $O(N)$ and each iteration of Dijkstra algorithm can be performed in $O(N^2)$ , while it is $O(N^3)$ for Bellman-Ford.

Implementation ¶

The implementation given here is long, it can probably be significantly reduced. It uses the SPFA algorithm for finding shortest paths.

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by clicking the link above. You may have to or before you can post: click the register link above to proceed. To start viewing messages, select the forum that you want to visit from the selection below. .--> 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. | , Posting Permissions post new threads post replies post attachments edit your posts is are code is code is

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