Karp's 21 NP-Complete Problems

Definition

Karp's 21 NP-Complete Problems are a set of decision problems that were identified by Richard Karp in 1972 as being both NP (nondeterministic polynomial time) and NP-complete. These problems are significant because they provide a framework for understanding the complexity of various computational problems and demonstrate that if any one of them can be solved in polynomial time, then every problem in NP can also be solved in polynomial time. Karp's work essentially established the foundation for the study of computational complexity.

5 Must Know Facts For Your Next Test

1. Karp identified 21 specific problems, including the Hamiltonian Cycle Problem and the Traveling Salesman Problem, which serve as classic examples of NP-complete problems.
2. These problems highlight the challenges in computer science regarding finding efficient algorithms, as no polynomial-time solutions have been found for any of them to date.
3. The concept of NP-completeness is central to computational theory and helps researchers classify problems based on their inherent difficulty.
4. Karp's work built on the foundation laid by Stephen Cook's theorem, which first established the concept of NP-completeness with the Boolean satisfiability problem (SAT).
5. If any one of Karp's 21 problems is proven to be solvable in polynomial time, it would imply that P = NP, fundamentally changing our understanding of computational limits.

Review Questions

• How do Karp's 21 NP-Complete Problems contribute to our understanding of computational complexity?
  • Karp's 21 NP-Complete Problems provide critical insights into computational complexity by categorizing difficult decision problems. They illustrate the nature of NP-completeness, showing that if one can find a polynomial-time solution for any of these problems, it would lead to polynomial-time solutions for all problems in NP. This classification helps researchers understand the relationships among various problems and guides efforts to develop efficient algorithms.
• Discuss how Karp's identification of these NP-complete problems relates to Cook's theorem and its implications.
  • Karp's identification of 21 NP-complete problems builds directly on Cook's theorem, which established that the Boolean satisfiability problem (SAT) is NP-complete. Karp used reductions to show that each of his identified problems is at least as hard as SAT, thus solidifying the concept that there exists a hierarchy within NP-complete problems. This connection emphasizes that solving any single NP-complete problem efficiently would solve all others in this class, making it pivotal for theoretical computer science.
• Evaluate the potential impact on computer science if a polynomial-time algorithm were discovered for one of Karp's NP-Complete Problems.
  • If a polynomial-time algorithm were found for one of Karp's NP-Complete Problems, it would have profound implications for computer science. It would imply that P = NP, revolutionizing our understanding of computational feasibility. This could lead to breakthroughs in various fields such as cryptography, optimization, and artificial intelligence since many practical problems are reducible to these NP-complete cases. The discovery would challenge existing theories and necessitate a reevaluation of algorithms and problem-solving strategies across numerous disciplines.

Related terms

NP-Complete: A class of decision problems for which no known polynomial-time algorithms exist, but if one problem can be solved quickly, all problems in NP can be solved quickly.

P vs NP Problem: An unsolved question in computer science that asks whether every problem whose solution can be quickly verified can also be quickly solved.

Reduction: A method for transforming one problem into another problem to show that if one problem can be solved, then the other can also be solved.