In our Design and Analysis Project, we came across a problem of which we have very little or no idea. The problem essestially breaks down to :

Given a rooted tree, find a minimum dominating set such that no two vertices in the dominating set share an edge and for all the vertices that is not in the dominating set, there exists exactly one vertex in the dominating set which is its neighbor. If such a set does not exists, report -1.

Any help as to how to approach the problem will be really appreciated.

Thank you.

The problem you’re facing is about finding a special set of vertices in a tree-like structure. This set is called a “dominating set.” The goal is to find the smallest dominating set possible. In this set, no two vertices share an edge, and for every vertex that is not in the set, there should be exactly one vertex in the set that is connected to it.

To approach this problem, you can follow these steps:

- Understand the problem: Make sure you fully understand the requirements and terms used in the problem. For example, a “rooted tree” is a structure where one vertex is designated as the root, and all other vertices have a unique path connecting them to the root.
- Define the problem: Clearly describe what the problem is asking for, including the input and desired output. Take note of any restrictions mentioned in the problem statement.
- Choose an algorithm: Decide on a method to solve the problem. There are different algorithms available, each with its own advantages and trade-offs. Some algorithms may provide exact solutions, while others may give approximations or estimates.
- Implement the algorithm: Write code in your preferred programming language to implement the chosen algorithm. This involves representing the input structure in a suitable way and writing the necessary logic to solve the problem.
- Test and validate: Create test cases to verify that your implementation works correctly. Test it with different inputs, including special cases and larger instances, to ensure it produces the expected results. Check if the output satisfies the requirements of a minimum dominating set.
- Optimize if needed: If your algorithm is not efficient enough for larger inputs, consider optimizing it. You can explore techniques like pruning (eliminating unnecessary computations), memoization (caching intermediate results), or using specialized data structures to improve performance.

It’s important to note that the Minimum Dominating Set problem can be quite challenging, especially for large structures. In such cases, finding an optimal solution may not be feasible within a reasonable time frame. In these situations, approximation algorithms or heuristics can provide good solutions that are close to optimal.

To gain a deeper understanding and explore more advanced techniques, you may want to refer to research papers, algorithms textbooks, or other resources that discuss the Minimum Dominating Set problem and related algorithms.

I hope this explanation helps you approach the problem in a more understandable way. Good luck with your project!