NP Problems

In the field of computational complexity, an NP problem (nondeterministic polynomial time) is characterized by each of its potential solutions being verifiable within polynomial time.

This implies that, although discovering a solution might be a time-consuming process, once proposed, its validity can be quickly ascertained.

However, this doesn't necessarily mean that there is an efficient algorithm available to find a solution.

In simpler terms, should a solution be presented, either by a guess or through a nondeterministic process, its correctness can be swiftly validated.

The difference between P and NP problems. The NP class encompasses both problems that have known polynomial-time algorithms (P problems) and those without known efficient solutions. Thus, P problems form a subset of the NP class. $$ P \subset NP $$ Moreover, NP includes NP-complete problems, for which a polynomial-time solution to any one would allow all NP problems to be resolved efficiently.

Example

To better understand, let's look at a well-known example: the Travelling Salesman Problem (TSP).

The task involves finding the shortest route that visits a list of cities and returns to the original city, knowing the distances between each city pair.

Although it’s easy to check if a particular route is feasible and to calculate its total distance, identifying the most efficient route becomes exceedingly difficult and time-intensive as the number of cities grows.

NP problems are crucial because they include a wide range of practical challenges in areas such as optimization, cryptography, and resource planning, among others.

While there are algorithms that can address these issues, many are not scalable due to the exponential increase in time required to solve larger instances of the problem.

The open question of P vs NP. A fundamental unresolved question in computer science is whether every problem that can be verified quickly (NP) can also be solved quickly (P). This is known as the P vs NP dilemma, and it remains one of the millennium's most