Richard Karp: Algorithms and Computational Complexity | Lex Fridman Podcast #111

NP-completeness

The concept of NP-completeness, introduced by Stephen Cook and Richard Karp, revolutionized algorithm theory by showing that many complex problems are interconnected. Karp explains that the satisfiability problem, a fundamental issue in computer science, can be expressed as any problem in NP, highlighting its universal nature 1. This means that if one NP-complete problem can be solved efficiently, all can be 2. He emphasizes the importance of measuring algorithm complexity by the worst-case scenario, as some algorithms may perform well generally but poorly in specific cases 3.

It's a nice statement of a really hard problem. And what Cook showed is that every problem in NP can be re-expressed as an instance of the satisfiability problem.

--- Richard Karp

This understanding underscores the significance of the P vs NP problem, which questions whether every problem whose solution can be quickly verified can also be solved quickly.

   

P vs NP

The P vs NP dilemma remains one of the most profound questions in theoretical computer science. Richard Karp argues that if P equals NP, it would mean that every problem with a verifiable solution could also be solved efficiently, a notion that seems unlikely given the historical difficulty of such problems 4. He believes that proving P equals NP or not will require new concepts beyond current understanding 5.

If P were equal to NP, it would be amazing. It would mean that every problem where a solution can be rapidly checked can actually be solved in polynomial time.

--- Richard Karp

The implications of solving this problem are vast, potentially transforming both theoretical computer science and practical software systems.

   

Polynomial Algorithms

Polynomial time algorithms are central to defining what is considered efficient in computational terms. Richard Karp explains that these algorithms have a running time that grows polynomially with input size, making them feasible for large-scale problems 6. He discusses the relationships among complexity classes, noting that breakthroughs in understanding these relationships could lead to significant advancements in computer science 7.

We theoreticians take that to be the definition of an algorithm being efficient, and we're interested in which problems can be solved by such efficient algorithms.

--- Richard Karp

Understanding these classes helps in identifying which problems can be solved efficiently and which require more innovative approaches.