Introduction to Computational Complexity Theory

Overview

Computational complexity theory studies the complexity of algorithms and problems. It aims to identify the computational resources required to solve a problem, such as time, space, or communication, and to determine the limitations and possibilities of algorithmic efficiency.

One of the most important concepts in computational complexity theory is the P versus NP problem, which asks whether every problem that can be verified in polynomial time can also be solved in polynomial time. The class P consists of problems that can be solved in polynomial time, while the class NP consists of problems for which a solution can be verified in polynomial time.

NP-complete problems are a special subset of NP problems that are believed to be among the hardest problems in the class, in the sense that any NP-complete problem can be reduced to any other NP-complete problem in polynomial time. The notion of NP-completeness has important implications in cryptography, optimization, and many other areas of computer science and mathematics.

Other important concepts in computational complexity theory include the notion of polynomial-time reductions, which allows the complexity of one problem to be compared to another, and the complexity classes beyond P and NP, such as PSPACE and EXPTIME.

Overall, computational complexity theory is a rich and important field of study that provides a framework for understanding the computational limits and possibilities of algorithms and problems.

Solvable vs. Unsolvable Problems

In computational complexity theory, problems are often classified as either solvable or unsolvable, based on whether there exists an algorithm that can solve the problem or not.

1. Solvable Problem - A problem is said to be solvable if there exists an algorithm that can solve it in a finite amount of time, i.e., for any input of the problem, the algorithm can produce the correct output in a finite number of steps. For example, the problem of finding the sum of two integers is solvable, since there exists a simple algorithm that can add two integers in a finite number of steps.
2. Unsolvable Problem - On the other hand, a problem is said to be unsolvable if there does not exist any algorithm that can solve it in a finite amount of time. One well-known example of an unsolvable problem is the Halting problem, which asks whether a given program will eventually halt or run forever. It has been proven that there is no algorithm that can solve the Halting problem for all possible inputs.

It is important to note that unsolvable problems are not the same as difficult problems. Some problems may be solvable, but require an exponential amount of time or space to solve, making them practically infeasible for large input sizes. The study of computational complexity theory focuses on understanding the limits of algorithmic efficiency, and identifying which problems are solvable, which problems are unsolvable, and which problems are solvable but infeasible.

Decidable vs. Undecidable Problems

In computational complexity theory, solvable problems are also often classified as decidable or undecidable, based on whether there exists an algorithm that can decide the problem or not.

1. Decidable Problem - A problem is said to be decidable if there exists an algorithm that can determine whether an instance of the problem is a "yes" instance or a "no" instance in a finite amount of time. In other words, the algorithm will always halt and return the correct answer, either "yes" or "no". For example, the problem of determining whether a given number is prime is decidable, since there exists an algorithm that can determine the primality of a number in a finite amount of time.
2. Undecidable Problem - On the other hand, a problem is said to be undecidable if there is no algorithm that can determine whether an instance of the problem is a "yes" instance or a "no" instance in a finite amount of time. The classic example of an undecidable problem is the Halting problem, which asks whether a given program will eventually halt or run forever. It has been proven that there is no algorithm that can solve the Halting problem for all possible inputs.

It is worth noting that undecidability is a stronger notion than insolvability. All undecidable problems are unsolvable, but not all unsolvable problems are undecidable. For example, the problem of finding an optimal solution to the traveling salesman problem is unsolvable for large input sizes, but it is still decidable, since there exist algorithms that can find a suboptimal solution in a finite amount of time.

The study of decidable and undecidable problems is important for understanding the limits of algorithmic solvability, and for identifying the scope and limits of automated decision-making in computer science and other fields.

P vs. NP Problems

In computational complexity theory, decidable problems are also often classified as P or NP problems.

The P versus NP problem is one of the most famous open problems in computer science and mathematics. It asks whether every problem that can be verified in polynomial time can also be solved in polynomial time. In other words, if a solution to a problem can be checked efficiently (in polynomial time), can it also be found efficiently (in polynomial time)?

The class of problems that can be solved in polynomial time is known as P, while the class of problems that can be verified in polynomial time is known as NP. Many important computational problems are known to be in NP, including the traveling salesman problem, the knapsack problem, and the satisfiability problem.

The P versus NP problem has important implications for cryptography, optimization, and other fields of computer science and mathematics. If P = NP, then it would be possible to solve many important computational problems much more efficiently than is currently known, and this could have major implications for fields such as cryptography and data analysis. However, if P ? NP, then many important computational problems may be fundamentally intractable, and this could have important implications for the limits of automated decision-making.

Despite decades of research, the P versus NP problem remains one of the most challenging open problems in computer science and mathematics, and it is not yet clear whether it will be resolved in the near future.

What are the major problems in Computational Complexity?

There are several major problems in computational complexity theory that remain open and actively researched. Here are a few examples:

1. P versus NP problem: As I mentioned earlier, this is one of the most famous open problems in computer science. It asks whether every problem that can be verified in polynomial time can also be solved in polynomial time.
2. Complexity of specific problems: There are many important computational problems for which the complexity is not yet well understood. Examples include the traveling salesman problem, graph isomorphism, and factoring large integers.
3. Circuit lower bounds: Despite many efforts, there is still no known general technique for proving lower bounds on the size of Boolean circuits that compute specific functions. This is a major open problem in computational complexity theory.
4. De-randomization: Randomized algorithms are often more efficient than deterministic algorithms, but it is not known whether every randomized algorithm can be converted into an equally efficient deterministic algorithm.
5. Quantum complexity theory: Quantum computers have the potential to solve certain problems much faster than classical computers, but the study of quantum complexity theory is still in its early stages.

These are just a few examples of the many open problems in computational complexity theory. Despite their difficulty, progress in these areas can have significant implications for cryptography, optimization, machine learning, and other fields of computer science and mathematics.