How can you identify NP-complete problems in your domain?

1 Recognize the problem class

The first step is to recognize the problem class of your task. Is it a decision problem, where you have to answer yes or no to a question? Is it an optimization problem, where you have to find the best solution among many alternatives? Is it a search problem, where you have to find a specific solution or a set of solutions? Decision problems are the easiest to analyze for NP-completeness, because they have a clear definition of correctness and a binary output. Optimization and search problems can be converted to decision problems by adding a threshold or a criterion.

2 Find a known NP-complete problem

The second step is to find a known NP-complete problem that is similar or related to your task. There are many NP-complete problems that have been proven by researchers, such as the traveling salesman problem, the satisfiability problem, the knapsack problem, and the vertex cover problem. You can look for these problems in textbooks, papers, or online resources. The idea is to find a problem that shares some features or constraints with your task, such as the input format, the output format, the objective function, or the combinatorial nature.

3 Apply a reduction technique

The third step is to apply a reduction technique to show that your task is at least as hard as the known NP-complete problem. A reduction technique is a way of transforming one problem into another problem in polynomial time, such that a solution to the second problem can be used to obtain a solution to the first problem. For example, you can reduce the satisfiability problem to the vertex cover problem by constructing a graph from a logical formula, and vice versa. By doing this, you can show that if there is an efficient algorithm for your task, then there is also an efficient algorithm for the known NP-complete problem, which is unlikely.

4 Check the correctness and complexity

The fourth step is to check the correctness and complexity of your reduction technique. You have to make sure that the transformation preserves the essential properties of the original problem, such as the input size, the output size, the answer, and the objective function. You also have to make sure that the transformation can be done in polynomial time, which means that the time required to perform the transformation grows at most as a polynomial function of the input size. If these conditions are met, then you have successfully shown that your task is NP-complete.

5 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?