View source for Richard Karp's 21 NP-complete problems

== Introduction ==

In the realm of computational complexity theory, the concept of [[NP-completeness]] plays a pivotal role in understanding the limits of efficient computation. Richard Karp, a prominent computer scientist, made a significant contribution to this field by identifying 21 NP-complete problems in his seminal 1972 paper titled "Reducibility Among Combinatorial Problems." This work not only expanded the list of known NP-complete problems but also provided a framework for demonstrating the NP-completeness of other problems. This article delves into the intricacies of these 21 problems, exploring their definitions, implications, and interconnections.

== Background on NP-Completeness ==

To comprehend the significance of Karp's 21 NP-complete problems, it is essential to understand the concept of NP-completeness itself. In computational complexity theory, the class [[NP]] (nondeterministic polynomial time) consists of decision problems for which a given solution can be verified in polynomial time. A problem is NP-complete if it is both in NP and as hard as any problem in NP, meaning that any NP problem can be reduced to it in polynomial time. The notion of NP-completeness was introduced by [[Stephen Cook]] in 1971, who demonstrated that the [[Boolean satisfiability problem]] (SAT) is NP-complete.

== The 21 NP-Complete Problems ==

=== 1. Satisfiability (SAT) ===

The Boolean satisfiability problem, commonly referred to as SAT, is the problem of determining if there exists an assignment of truth values to variables that makes a given Boolean formula true. SAT was the first problem proven to be NP-complete and serves as a cornerstone for the theory of NP-completeness.

=== 2. 0/1 Integer Programming ===

0/1 Integer Programming involves finding a binary vector that satisfies a set of linear inequalities. This problem is a special case of the more general integer programming problem and is NP-complete due to its combinatorial nature.

=== 3. Clique ===

The Clique problem asks whether a given graph contains a complete subgraph (clique) of a specified size. This problem is fundamental in graph theory and has applications in social network analysis and bioinformatics.

=== 4. Vertex Cover ===

The Vertex Cover problem involves determining whether a graph contains a subset of vertices such that each edge in the graph is incident to at least one vertex in the subset. This problem is crucial in network security and resource allocation.

=== 5. Hamiltonian Cycle ===

The Hamiltonian Cycle problem asks whether a given graph contains a cycle that visits each vertex exactly once. This problem is significant in routing and scheduling applications.

=== 6. Hamiltonian Path ===

Similar to the Hamiltonian Cycle, the Hamiltonian Path problem seeks a path that visits each vertex exactly once without forming a cycle. This problem is applicable in various logistical and optimization scenarios.

=== 7. Traveling Salesman Problem (TSP) ===

The Traveling Salesman Problem involves finding the shortest possible route that visits a set of cities and returns to the origin city. TSP is a classic example of an NP-complete problem with numerous practical applications in logistics and planning.

=== 8. Subset Sum ===

The Subset Sum problem asks whether there exists a subset of a given set of integers that sums to a specified target value. This problem is a special case of the [[knapsack problem]] and is relevant in cryptography and resource allocation.

=== 9. Knapsack Problem ===

The Knapsack Problem involves selecting a subset of items with given weights and values to maximize the total value without exceeding a weight limit. This problem has applications in finance and resource management.

=== 10. Set Cover ===

The Set Cover problem asks whether a collection of sets can cover a universe of elements with a specified number of sets. This problem is vital in fields such as database systems and network design.

=== 11. Feedback Node Set ===

The Feedback Node Set problem involves finding a minimum set of vertices in a directed graph whose removal results in a graph without cycles. This problem is significant in circuit design and network analysis.

=== 12. Feedback Arc Set ===

The Feedback Arc Set problem requires finding a minimum set of edges in a directed graph whose removal results in a graph without cycles. This problem has applications in scheduling and ranking systems.

=== 13. Directed Hamiltonian Cycle ===

The Directed Hamiltonian Cycle problem is a variant of the Hamiltonian Cycle problem, where the graph is directed. This problem is crucial in analyzing directed networks and circuits.

=== 14. Directed Hamiltonian Path ===

Similar to the Directed Hamiltonian Cycle, the Directed Hamiltonian Path problem seeks a path in a directed graph that visits each vertex exactly once. This problem is applicable in various directed network scenarios.

=== 15. Exact Cover by 3-Sets (X3C) ===

The Exact Cover by 3-Sets problem involves determining whether a collection of 3-element subsets can cover a universe of elements exactly once. This problem is a special case of the set cover problem and is relevant in combinatorial design.

=== 16. 3-SAT ===

3-SAT is a specific form of the SAT problem, where each clause in the Boolean formula has exactly three literals. This problem is widely studied in theoretical computer science and has implications in logic and circuit design.

=== 17. 3-Dimensional Matching ===

The 3-Dimensional Matching problem involves finding a perfect matching in a tripartite hypergraph. This problem is significant in database systems and resource allocation.

=== 18. Partition ===

The Partition problem asks whether a given set of integers can be partitioned into two subsets with equal sums. This problem is a special case of the subset sum problem and is relevant in load balancing and resource allocation.

=== 19. Job Sequencing ===

The Job Sequencing problem involves scheduling jobs on a single machine to minimize the total weighted completion time. This problem is crucial in operations research and production planning.

=== 20. Bin Packing ===

The Bin Packing problem involves packing objects of different sizes into a finite number of bins with a fixed capacity. This problem has applications in logistics and resource management.

=== 21. Graph Coloring ===

The Graph Coloring problem asks whether a given graph can be colored with a specified number of colors such that no two adjacent vertices share the same color. This problem is fundamental in scheduling and frequency assignment.

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== Implications and Applications ==

The identification of these 21 NP-complete problems by Richard Karp has profound implications for both theoretical and practical aspects of computer science. These problems serve as benchmarks for evaluating the complexity of new problems and provide a foundation for developing approximation algorithms and heuristics. In practice, NP-complete problems arise in diverse fields such as cryptography, operations research, artificial intelligence, and bioinformatics.

== Techniques for Proving NP-Completeness ==

Karp's methodology for proving NP-completeness involves demonstrating a polynomial-time reduction from a known NP-complete problem to the problem in question. This technique, known as a [[Karp reduction]], is a powerful tool for establishing the complexity of new problems. By leveraging the transitive property of polynomial-time reductions, researchers can build a network of NP-complete problems, facilitating the identification of new members of this class.

== Challenges and Open Questions ==

Despite the extensive study of NP-complete problems, several challenges and open questions remain. The most prominent of these is the [[P vs NP problem]], which asks whether every problem in NP can be solved in polynomial time. A resolution to this question would have far-reaching implications for computer science, mathematics, and related fields. Additionally, the development of efficient approximation algorithms for NP-complete problems continues to be an active area of research, with the goal of finding near-optimal solutions in a reasonable timeframe.

== See Also ==

* [[Computational Complexity Theory]]
* [[Approximation Algorithms]]
* [[P vs NP Problem]]

[[Category:Computational Complexity]]
[[Category:Computer Science]]
[[Category:Combinatorial Optimization]]