P versus NP problem

Introduction

The P versus NP problem is a fundamental question in computer science, specifically in the field of computational complexity theory. It asks whether every problem for which a solution can be quickly checked by a computer can also be quickly solved by a computer. The terms "quickly checked" and "quickly solved" are quantified using the concept of polynomial time.

A computer with mathematical equations on the screen, symbolizing the complexity of the P versus NP problem.

Definitions

Before diving deeper into the P versus NP problem, it is crucial to understand some key terms and concepts.

Polynomial Time

In computer science, an algorithm is said to run in polynomial time if its running time is upper bounded by a polynomial expression in the size of the input for the algorithm. In other words, the running time grows at most as some fixed power of the size of the input. Polynomial time is often used as a synonym for "efficiently solvable".

P Class

The class P consists of those decision problems that can be solved by a deterministic Turing machine within polynomial time, or equivalently, by a deterministic algorithm within polynomial time.

NP Class

The class NP consists of those decision problems for which a given solution can be verified as correct or incorrect in polynomial time by a deterministic Turing machine, or equivalently, by a deterministic algorithm.

The Problem

The P versus NP problem, posed by Stephen Cook in 1971 and independently by Leonid Levin in 1973, asks whether P equals NP. In other words, it asks whether every problem for which a solution can be quickly checked can also be quickly solved. If P does equal NP, it would mean that there is a fast (polynomial time) algorithm for every problem for which a solution can be quickly checked. If P does not equal NP, it would mean that there are problems for which solutions can be quickly checked but not quickly found.

Implications

The implications of the P versus NP problem are far-reaching. If P equals NP, then the world would be radically different. Many tasks that are currently computationally infeasible, such as factoring large numbers or solving the traveling salesman problem, would become feasible. This would have profound implications for fields such as cryptography, operations research, database theory, AI, and more.

On the other hand, if P does not equal NP, then there are problems that are "hard" in the sense that they cannot be quickly solved. This would confirm the intuition of many computer scientists that there are inherent limitations to what can be computed efficiently.

Current Status

As of now, the P versus NP problem remains unsolved. It is one of the seven Millennium Prize Problems for which the Clay Mathematics Institute offers a $1 million prize for a correct solution. Despite the efforts of many brilliant minds, we are no closer to a definitive answer than we were when the problem was first posed.