Karp's 21 NP-complete problems

Last updated April 26, 2023

In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete. In his 1972 paper, "Reducibility Among Combinatorial Problems", ^[1] Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete^[2] (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction from the boolean satisfiability problem to each of 21 combinatorial and graph theoretical computational problems, thereby showing that they are all NP-complete. This was one of the first demonstrations that many natural computational problems occurring throughout computer science are computationally intractable, and it drove interest in the study of NP-completeness and the P versus NP problem.

The problems

Karp's 21 problems are shown below, many with their original names. The nesting indicates the direction of the reductions used. For example, Knapsack was shown to be NP-complete by reducing Exact cover to Knapsack.

Satisfiability : the boolean satisfiability problem for formulas in conjunctive normal form (often referred to as SAT)
- 0–1 integer programming (A variation in which only the restrictions must be satisfied, with no optimization)
- Clique (see also independent set problem)
  - Set packing
  - Vertex cover
    - Set covering
    - Feedback node set
    - Feedback arc set
    - Directed Hamilton circuit (Karp's name, now usually called Directed Hamiltonian cycle)
      - Undirected Hamilton circuit (Karp's name, now usually called Undirected Hamiltonian cycle)
- Satisfiability with at most 3 literals per clause (equivalent to 3-SAT)
  - Chromatic number (also called the Graph Coloring Problem)
    - Clique cover
    - Exact cover
      - Hitting set
      - Steiner tree
      - 3-dimensional matching
      - Knapsack (Karp's definition of Knapsack is closer to Subset sum)
        Job sequencing
        Partition
        Max cut

Approximations

As time went on it was discovered that many of the problems can be solved efficiently if restricted to special cases, or can be solved within any fixed percentage of the optimal result. However, David Zuckerman showed in 1996 that every one of these 21 problems has a constrained optimization version that is impossible to approximate within any constant factor unless P = NP, by showing that Karp's approach to reduction generalizes to a specific type of approximability reduction.^[3] Note however that these may be different from the standard optimization versions of the problems, which may have approximation algorithms (as in the case of maximum cut).

Notes

Related Research Articles

In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.

The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved.

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.

In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length and if every other problem that can be solved in polynomial space can be transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, the class of decision problems solvable in polynomial space, because a solution to any one such problem could easily be used to solve any other problem in PSPACE.

In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph. Both problems are NP-complete.

In computer science, the clique problem is the computational problem of finding cliques in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique, finding a maximum weight clique in a weighted graph, listing all maximal cliques, and solving the decision problem of testing whether a graph contains a clique larger than a given size.

Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.

In the mathematical area of graph theory, a clique is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph $is an induced subgraph of that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.$

In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem.

In the mathematical discipline of graph theory, a feedback vertex set (FVS) of a graph is a set of vertices whose removal leaves a graph without cycles. Equivalently, each FVS contains at least one vertex of any cycle in the graph. The feedback vertex set number of a graph is the size of a smallest feedback vertex set. The minimum feedback vertex set problem is an NP-complete problem; it was among the first problems shown to be NP-complete. It has wide applications in operating systems, database systems, and VLSI chip design.

In computational complexity theory, the class APX is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant. In simple terms, problems in this class have efficient algorithms that can find an answer within some fixed multiplicative factor of the optimal answer.

In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut. In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions.

In computational complexity theory, the PCP theorem states that every decision problem in the NP complexity class has probabilistically checkable proofs of constant query complexity and logarithmic randomness complexity.

In computational complexity theory, a gadget is a subset of a problem instance that simulates the behavior of one of the fundamental units of a different computational problem. Gadgets are typically used to construct reductions from one computational problem to another, as part of proofs of NP-completeness or other types of computational hardness. The component design technique is a method for constructing reductions by using gadgets.

In graph theory, a clique cover or partition into cliques of a given undirected graph is a partition of the vertices into cliques, subsets of vertices within which every two vertices are adjacent. A minimum clique cover is a clique cover that uses as few cliques as possible. The minimum $k$ for which a clique cover exists is called the clique cover number of the given graph.

In computational complexity theory, a problem is NP-complete when:

It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no".
When the answer is "yes", this can be demonstrated through the existence of a short solution.
The correctness of each solution can be verified quickly and a brute-force search algorithm can find a solution by trying all possible solutions.
The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified.

In computational complexity theory, the exponential time hypothesis is an unproven computational hardness assumption that was formulated by Impagliazzo & Paturi (1999). It states that satisfiability of 3-CNF Boolean formulas cannot be solved in subexponential time, i.e., $for all constant, where n is the number of variables in the formula. The exponential time hypothesis, if true, would imply that P \neq NP, but it is a stronger statement. It implies that many computational problems are equivalent in complexity, in the sense that if one of them has a subexponential time algorithm then they all do, and that many known algorithms for these problems have optimal or near-optimal time complexity.$

References

Cook, Stephen (1971). "The Complexity of Theorem Proving Procedures". Proc. 3rd Annual ACM Symposium on Theory of Computing (STOC). pp. 151–158. doi:10.1145/800157.805047. ISBN 9781450374644. S2CID 7573663.
Karp, Richard M. (1972). "Reducibility Among Combinatorial Problems" (PDF). In R. E. Miller; J. W. Thatcher; J.D. Bohlinger (eds.). Complexity of Computer Computations. New York: Plenum. pp. 85–103. doi:10.1007/978-1-4684-2001-2_9. ISBN 978-1-4684-2003-6.
Zuckerman, David (1996). "On Unapproximable Versions of NP-Complete Problems". SIAM Journal on Computing. 25 (6): 1293–1304. doi:10.1137/S0097539794266407.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[FOOTNOTEKarp1972-1] Karp 1972.

[FOOTNOTECook1971-2] Cook 1971.

[FOOTNOTEZuckerman1996-3] Zuckerman 1996.

[1]

[2]

[3]