Computational complexity theory - Wikipedia

Computational Complexity Theory Overview

• Core Definition: A branch of theoretical computer science that classifies computational problems based on the resources (time, storage, communication, etc.) required to solve them.
• Key Objective: To determine the practical limits of what computers can and cannot do by analyzing all possible algorithms for a given problem, rather than just one specific algorithm.
• Problem vs. Instance:
  • Problem: An abstract question (e.g., primality testing).
  • Instance: A specific input for that problem (e.g., the number 15). Complexity theory focuses on the problem, not individual instances.
• Types of Problems:
  • Decision Problems: Problems with a "yes" or "no" output (e.g., "Is this graph connected?"). These are the central objects of study and can be viewed as formal languages.
  • Function Problems: Problems requiring a more complex output (e.g., integer factorization or the traveling salesman problem). These can be recast as decision problems.
• Measuring Complexity:
  • Complexity is expressed as a function of the input size ($n$), typically measured in bits.
  • Worst-case time complexity: The maximum time required for any input of size $n$.
  • Polynomial time: Algorithms where the time taken is a polynomial function of $n$.
  • Cobham's Thesis: Proposes that problems solvable in polynomial time are "feasible."
• Mathematical Models:
  • Turing Machine: The standard theoretical model used to represent any computing device.
  • Church–Turing Thesis: The belief that any algorithmically solvable problem can be solved by a Turing machine.
  • Machine Variations: Includes deterministic, probabilistic, non-deterministic, and quantum Turing machines, which help define various complexity classes.
• Related Fields:
  • Analysis of Algorithms: Focuses on the resource usage of a specific algorithm.
  • Computability Theory: Asks what can be solved in principle, without resource restrictions.
• Notable Problem: The P versus NP problem, one of the seven Millennium Prize Problems, is a central topic in the field.