Computational Complexity Theory

Computational complexity theory is a specialized branch of theoretical computer science dedicated to examining the inherent difficulty of various computational problems.

It hinges on a pivotal question: why are certain problems computationally challenging while others are not?

The field explores the nature of these problems and seeks to establish a systematic and quantifiable framework to compare and categorize their complexity.

It aims to delineate the limits of practical computation—not merely theoretical—by evaluating the resources, such as time and memory, required to solve specific problems.

This field is distinct from computability theory, which primarily focuses on whether a problem can be computed without considering the necessary resources.

Historical Origins and Developments

Emerging as a distinct discipline within computer science during the 1960s and '70s, computational complexity theory was shaped by the work of pioneers like Jack Edmonds, Stephen Cook, and Leonid Levin. One of the field’s earliest and most influential achievements was Cook’s formalization of the NP-complete class of problems in 1971, which initiated extensive research into computational difficulty and the interrelations between different problem classes.

Principal Complexity Classes

One of the significant advancements in complexity theory has been the development of classification schemes that organize computational problems based on their difficulty levels.

This system enables scientists to pinpoint and assess the computational difficulty of particular problems, even when direct proof of their complexity is unattainable.

These schemes resemble the periodic table of elements, where problems are categorized by their computational "weight" in a manner akin to how elements are sorted by atomic number and chemical properties, starting from the lightest element, hydrogen, to the heaviest.

In particular, complexity theory categorizes computational problems into various classes based on the resources they require for resolution. The most studied classes include:

• P: This class includes problems that can be solved in polynomial time by a deterministic Turing machine. It represents problems considered "easily" solvable.
• NP: Comprising problems whose solutions can be verified in polynomial time by a deterministic Turing machine. It includes all problems in P and possibly others that are not solvable in polynomial time.
• NP-Complete: This subclass contains the most challenging problems within NP, indicating that if one can be solved in polynomial time, then all problems in NP can be likewise resolved.
• Co-NP: Consists of problems whose complements fall within NP, meaning that for these issues, a negative solution can be quickly verified.
• PSPACE: Includes problems that can be resolved with a polynomial amount of memory, regardless of the time required. This class is broader than P and NP and includes problems potentially requiring exponential time to solve.
• EXPTIME: Contains problems solvable in exponential time relative to input size, representing a significant escalation in difficulty compared to those in P or NP.
• SPACE: Similar to PSPACE, but with no restriction to polynomial limits. It includes problems solvable with any amount of space.
• NC: Includes problems that can be efficiently solved using a polynomial number of processors in polynomial time on a parallel computing machine.
• AC: A subclass of NC, encompassing problems solvable with Boolean circuits of logarithmic depth and gates that can have numerous inputs.
• BPP: Stands for Bounded Error Probabilistic Polynomial time, including problems that can be solved probabilistically in polynomial time with a likelihood of error less than one-third for any given input.
• #P: Encompasses counting problems related to decision problems in NP, involving the counting of the number of solutions to a problem within NP.
• L: Comprises problems solvable with a logarithmic amount of memory, representing a very restricted subclass compared to PSPACE.
• EXPSPACE: Similar to EXPTIME in its spatial complexity, including problems solvable using an exponential amount of space.

These complexity classes help researchers understand the structure and interrelationships of computational problems, aiding in the development of efficient algorithms and the delineation of computing’s theoretical limits.

Typically, the easiest route, which consumes the least amount of memory and processing time, is sought to solve a computational problem. Cryptography is an exception where computationally "difficult" problems are advantageous for ensuring robust data protection and security. For example, a strong encryption code requires significant computational time and resources to crack.

Open Questions and Current Research

Whether P equals NP is one of the most renowned open questions in mathematics and computer science, with profound implications for complexity theory, cryptography, optimization, and several other fields. Despite decades of research, this question remains unresolved.

Modern research in complexity theory also ventures beyond traditional classes like P and NP, exploring new computational models, such as quantum computing, and novel complexity classes. These efforts strive to deepen our understanding of computation’s nature and forge new strategies for tackling complex computational challenges.

In conclusion, computational complexity theory remains a vibrant and essential area of study within computer science, offering key insights into the boundaries of computation and steering the development of efficient algorithms. With its unresolved questions and ongoing advancements, it continues to hold a central place in scientific inquiry for the foreseeable future.