Karp's Paper: Insights on NP-Completeness
In 1972, Richard Karp's paper (Reducibility in Combinatorial Problems) emerged, which is not only a milestone in computer science but also a profound exploration of the nature of computation.
Gödel and Recursive Functions
As early as the 1930s, Gödel introduced the concept of recursive functions. A recursive function is a function that can call itself, and it has important applications in both mathematics and computer science. Gödel proved that there are some recursive functions for which we cannot determine whether they will halt in a finite amount of time.
Turing and Computability
Meanwhile, Turing proposed the model of the Turing machine. A Turing machine is an abstract computational device that can simulate any computational process. Turing proved that there are some problems for which a Turing machine cannot provide an answer in a finite amount of time, thus there exist uncomputable problems.
Karp and NP-Completeness
Karp's paper linked Gödel's recursive functions and Turing's theory of computability with practical problems. He proved that a series of important combinatorial problems, such as the traveling salesman problem and the set cover problem, all belong to NP-complete problems. NP-complete problems refer to the situation where if there exists a polynomial time algorithm that can solve any one of these problems, then all NP problems can be solved by a polynomial time algorithm.
However, to this day, we have not found any polynomial time algorithm for any NP-complete problem, and whether P equals NP remains the greatest unsolved mystery in computer science.
Summary
Karp's paper revealed the limitations of computation. There are some problems for which we can verify the correctness of a solution, but it is difficult to find the solution itself. This is not only a challenge for computer science but also a challenge to human cognitive abilities. Karp's work not only advanced the development of computational complexity theory but also sparked deep reflections on the nature of computation, knowledge, and the world.