Computational complexity theory is a field within computer science that aims to classify and study the complexity of solving computational problems. It categorizes problems based on how their difficulty scales with the size of the input. The most common classification frameworks used in computational complexity theory are the distinction between P, NP, and NP-hard problems.

  1. P (Polynomial Time): These are problems that can be solved in polynomial time, which means that the algorithm's running time is bounded by a polynomial function of the input size. P problems are considered "efficient" and can be solved in a reasonable amount of time for large inputs. Examples of P problems include sorting an array, finding the shortest path in a graph, and matrix multiplication.

  2. NP (Nondeterministic Polynomial Time): NP is a set of problems for which a proposed solution can be verified in polynomial time. In other words, given a potential solution, it can be checked in polynomial time to determine if it is correct. NP problems are generally harder to solve than P problems, and no efficient algorithm is known for solving them. Examples of NP problems include the traveling salesman problem and the Boolean satisfiability problem (SAT).

  3. NP-hard: NP-hard problems are at least as hard as the hardest problems in NP. That is, if any NP-hard problem can be solved in polynomial time, then all problems in NP would also be solvable in polynomial time, effectively collapsing the distinction between P and NP. The classic example of an NP-hard problem is the Boolean satisfiability problem (SAT). NP-hard problems need not be in NP and may not have efficient solutions.

  In addition to these classifications, there is also the concept of NP-complete problems. NP-complete problems are a subset of NP problems that are both in NP and NP-hard. They are considered to be the most difficult problems in NP, and if any NP-complete problem has a polynomial-time algorithm, then all NP problems do.

  It is worth noting that computational complexity theory is concerned with the worst-case complexity of algorithms for solving problems. While a problem may have efficient solutions on average or in practice, computational complexity theory studies the inherent complexity of problems in the worst-case scenario.