NP-completeness

Introduction

In the field of computer science, NP-completeness is a fundamental concept related to computational complexity theory. It is a property of certain decision problems and is crucial in understanding the limits of what computers can and cannot do.

Definition

A decision problem is NP-complete when it meets two criteria:

1. It is in NP (nondeterministic polynomial time), which means that given a "yes" instance of the problem, a solution can be verified in polynomial time.
2. Any problem in NP can be reduced to it using a polynomial time reduction.

The concept of NP-completeness was introduced in 1971 by Stephen Cook in his seminal paper "The Complexity of Theorem-Proving Procedures". Shortly after, Richard Karp demonstrated its applicability by showing that many other problems in computer science are also NP-complete.

Significance of NP-completeness

The significance of NP-completeness lies in its implications for solving decision problems. If a polynomial time algorithm can be found for one NP-complete problem, then a polynomial time algorithm can be found for all problems in NP. This is known as the P vs NP problem, one of the most important open questions in computer science.

Examples of NP-complete problems

There are many known NP-complete problems. Some of the most well-known include:

1. Travelling Salesman Problem (TSP): Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?
2. Knapsack Problem: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.
3. Boolean Satisfiability Problem (SAT): Given a Boolean expression, is there a true/false assignment to the variables that satisfies the expression?

Proving NP-completeness

To prove that a problem is NP-complete, it must be shown that the problem is in NP and that it is NP-hard. The most common method for proving NP-completeness involves reduction from another problem that is already known to be NP-complete. This is typically done by transforming instances of the known NP-complete problem into instances of the problem being considered.

Implications of NP-completeness

The existence of NP-complete problems has significant implications for computer science and related fields. If it can be shown that P ≠ NP, it would mean that there are problems that are fundamentally intractable, with no efficient solutions. Conversely, if P = NP, it would mean that every problem for which a solution can be verified quickly also has a solution that can be found quickly. This would have profound implications for fields such as cryptography, where the security of many systems relies on certain problems being difficult to solve.

See Also

• P vs NP Problem
• Computational Complexity Theory
• Polynomial Time Reduction