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Complexity class
computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that any problem in co-NP can be reformulated as
Co-NP-complete
Complexity class
NP-complete problems are the hardest of the problems to which solutions can be verified quickly. Somewhat more precisely, a problem is NP-complete when:
NP-completeness
Complexity class
{co-NP}}={\textsf {NP}}} . The proof that no co-NP-complete problem can be in NP if NP ≠ co-NP {\displaystyle {\textsf {NP}}\neq {\textsf {co-NP}}} is symmetrical
Co-NP
Unsolved problem in computer science
John von Neumann, Gödel asked whether theorem-proving (now known to be co-NP-complete) could be solved in quadratic or linear time, and posited that if so
P_versus_NP_problem
Complexity class used to classify decision problems
equal to NP, then a polynomial-time algorithm would exist for solving NP-complete, and by corollary, all NP problems. The complexity class NP is related
NP_(complexity)
Computational property
computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem
Strong_NP-completeness
Topics referred to by the same term
computational complexity class NP-complete, a class of decision problems NP-hard, a class of problems in computational complexity Co-NP, a complexity class Numpy
NP
Puzzle video game genre
to be consistent, solving it is not known to be NP-complete, but it has been proven to be co-NP-complete. In the latter case, however, minesweeper exhibits
Minesweeper_(video_game)
Australian computer scientist (born 1944)
testing problem in the complexity class NP and providing the first strong evidence that the problem is not co-NP-complete. The Knuth–Morris–Pratt algorithm
Vaughan_Pratt
Complexity class of problems
problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such problems
NP-intermediate
is co-NP-complete. That is, the decision problem whose answer is "yes" for a graph that is not 1-tough, and "no" for a graph that is 1-tough, is NP-complete
Graph_toughness
3-regular graph with no 3-edge-coloring
cubic graph is 3-edge-colorable is NP-complete. Therefore, determining whether a graph is a snark is co-NP-complete. W. T. Tutte conjectured that every
Snark_(graph_theory)
Criterion for evaluating fairness of electoral systems
time; PJR and EJR are coNP-complete to verify; PER is NP-hard to verify (deciding whether a perfect representation exists is NP-complete). The satisfaction
Justified_representation
Problem in combinatorial optimization
would mean that there is no solution with a larger V). This problem is co-NP-complete. There is a pseudo-polynomial time algorithm using dynamic programming
Knapsack_problem
Geometric property of a pair of sets of points in Euclidean geometry
2 n − O ( n ) {\displaystyle 2^{n^{2}-n\log _{2}n-O(n)}} . It is co-NP-complete to decide whether a Boolean function given in disjunctive or conjunctive
Linear_separability
Marker used in SQL databases to indicate a value does not exist
the problem whether a c-table represents some concrete relation has a co-NP-complete complexity, thus is of little practical worth. A weaker notion of representation
Null_(SQL)
Computer science concept
hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy
Polynomial_hierarchy
Problem of determining if a Boolean formula could be made true
problem that was proven to be NP-complete—this is the Cook–Levin theorem. This means that all problems in the complexity class NP, which includes a wide range
Boolean satisfiability problem
Boolean_satisfiability_problem
System of logic in mathematics and philosophy
NP-complete (this is a generalisation of Cook's theorem for classical propositional logic as pointed out in ). Therefore, the validity problem is co-NP
Łukasiewicz_logic
1979 classic textbook on computational complexity theory
Theory of NP-Completeness is a textbook by Michael Garey and David S. Johnson. It was the first book exclusively on the theory of NP-completeness and computational
Computers_and_Intractability
In logic, a statement which is always true
satisfiability problem is NP-complete, and consequently, tautology is co-NP-complete. It is widely believed that (equivalently for all NP-complete problems) no polynomial-time
Tautology_(logic)
Notion of the "hardest" or "most general" problem in a complexity class
known complete problems, whereas classes that lack a computable enumeration have none. For example, NP, co-NP, PLS, PPA all have known natural complete problems
Complete_(complexity)
Logical formula with NOT only on variables
the satisfiability problem continues to be NP-complete, and the validity problem continues to be co-NP-complete. For formulas in conjunctive normal form
Negation_normal_form
Decomposition of a number into a product
all three of the complexity classes P, NP-complete, and co-NP-complete. It is therefore a candidate for the NP-intermediate complexity class. In contrast
Integer_factorization
homomorphically equivalent. It is NP-complete to test whether a graph has a homomorphism to a proper subgraph, and co-NP-complete to test whether a graph is
Core_(graph_theory)
Computational problem
and its complement, the propositional tautology problem, which is complete for co-NP. Circuit satisfiability Switching lemma Samuel R. Buss (Jan 1987)
Circuit_value_problem
language L is in NP then the complement of L is in co-NP. (This does not mean that the complement of NP is co-NP—there are languages which are known to be in
List_of_complexity_classes
Maximum number of colors in a greedy graph coloring
equals its chromatic number. Testing whether a graph is well-colored is coNP-complete. The hereditarily well-colored graphs (graphs for which every induced
Grundy_number
{\displaystyle f(x,y)} is true. NP/poly is used in a variation of Mahaney's theorem on the non-existence of sparse NP-complete languages. Mahaney's theorem
NP/poly
Classic NP-complete problem in computer science
problem (SAT), and likewise, has been proven to be NP-complete. It is a prototypical NP-complete problem; the Cook–Levin theorem is sometimes proved
Circuit satisfiability problem
Circuit_satisfiability_problem
Existence of values making formula true
trivial, as every formula is satisfiable, while the validity problem is co-NP complete. In the case of classical propositional logic, satisfiability is decidable
Satisfiability
Abstract devices built up of a fixed number of "wires"
remain difficult for networks of large sizes, due to the problem being co-NP-complete. Knuth, D. E. (1997). The Art of Computer Programming, Volume 3: Sorting
Sorting_network
Subfield of automated reasoning and mathematical logic
the common case of propositional logic, the problem is decidable but co-NP-complete, and hence only exponential-time algorithms are believed to exist for
Automated_theorem_proving
Unsolved problem in computational complexity theory
solvable in polynomial time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. It is known that the graph
Graph_isomorphism_problem
Data structure for Boolean functions
the BDD of a Boolean function solves the NP-complete Boolean satisfiability problem and the co-NP-complete tautology problem, constructing the BDD can
Binary_decision_diagram
One-by-one assignment of colors to graph vertices
graphs in which all induced subgraphs are well-colored. However, it is co-NP-complete to determine whether a graph is well-colored. If a random graph is drawn
Greedy_coloring
Yes/no problem in computer science
the complexity of the characteristic functions of an NP-complete problem and its co-NP-complete complement is exactly the same even though the underlying
Decision_problem
Grundy number problem, it is NP-complete to test whether these two orderings exist. It follows that it is co-NP-complete to test whether a given graph
Well-colored_graph
Graph with equal-size maximal independent sets
time, but testing whether other kinds of graph are well-covered is a coNP-complete problem. A vertex cover in an undirected graph is a set of vertices
Well-covered_graph
Surname list
of the SAT-problem; testing if a formula is a tautology, known to be co-NP-complete. This page lists people with the surname Taut. If an internal link intending
Taut
Problem of finding the longest simple path for a given graph
problem is NP-hard and the decision version of the problem, which asks whether a path exists of at least some given length, is NP-complete. This means
Longest_path_problem
Inherent difficulty of computational problems
{\textsf {NP}}} -complete, the polynomial time hierarchy will collapse to its first level (i.e., NP {\displaystyle {\textsf {NP}}} will equal co-NP {\displaystyle
Computational complexity theory
Computational_complexity_theory
global connectivity of the single-vertex recoloring state space is co-NP-complete, but when two colorings can be reconfigured to each other, the shortest
Reconfiguration
Matrix in linear algebra
semidefinite. The problem of deciding whether a matrix is copositive is co-NP-complete. Berman, Abraham; Robert J. Plemmons (1979). Nonnegative Matrices in
Copositive_matrix
When a finite set S of relations yields polynomial-time or NP-complete problems
set S of relations over the Boolean domain yields polynomial-time or NP-complete problems when the relations of S are used to constrain some of the propositional
Schaefer's_dichotomy_theorem
Field in logic and theoretical computer science
proof systems can be viewed as a step towards separating NP from co-NP (and thus P from NP), since the existence of a propositional proof system that
Proof_complexity
Theorem in order and lattice theory
Bronisław Knaster and Alfred Tarski, states the following: Let (L, ≤) be a complete lattice and let f : L → L be an order-preserving (monotonic) function with
Knaster–Tarski_theorem
On short connecting nets with added points
given threshold, is NP-complete, which implies that the optimization variant, asking for the minimum-weight tree in a given graph, is NP-hard. In fact, the
Steiner_tree_problem
language is co-NP-complete, then P = NP. (Mahaney, 1982) used this to prove Mahaney's theorem that if any sparse language is NP-complete, then P = NP. Mahaney's
Sparse_language
Version of classical propositional calculus that uses only one connective
the implicational propositional calculus is NP-complete, meaning that validity (tautology) is co-NP-complete. In this case, a useful technique is to presume
Implicational propositional calculus
Implicational_propositional_calculus
Class of problems in computer science
GISMPk is NP-complete even when k ≥ 2 {\displaystyle k\geq 2} . Moreover, GISMPk is MaxSNP-complete, i.e., it does not have a PTAS unless P=NP. This can
Interval_scheduling
Task of computing complete subgraphs
clique problem are hard. The clique decision problem is NP-complete (one of Karp's 21 NP-complete problems). The problem of finding the maximum clique is
Clique_problem
Fewest edge crossings in drawing of a graph
ISSN 0022-0000. Garey, M. R.; Johnson, D. S. (1983). "Crossing number is NP-complete". SIAM Journal on Algebraic and Discrete Methods. 4 (3): 312–316. doi:10
Crossing number (graph theory)
Crossing_number_(graph_theory)
Deciding whether a given allocation is 1-of- n {\displaystyle n} MMS is co-NP complete for agents with additive preferences. What is the largest fraction r
List of unsolved problems in fair division
List_of_unsolved_problems_in_fair_division
that NP is unequal to co-NP (the class of complements of languages in NP), which would imply more strongly that the complements of all NP-complete languages
Polynomial_creativity
Algorithm characteristic in computations
factorization and discrete log problems are in NP ∩ coNP, and are therefore not believed to be NP-complete. The fact that all of cryptography is predicated
Average-case_complexity
repetitive path is in NP, so testing whether a coloring is nonrepetitive is in co-NP, and Manin showed that it is co-NP-complete. The problem of finding
Thue_number
Logic puzzle
it means the cell that was probed for blackness must be white. It is NP-complete to solve Nurikabe, even when the involved numbers are 1 and 2 only. Further
Nurikabe_(puzzle)
Reduction between decision problems that preserves strong NP-completeness
kind of reduction between decision problems that preserves strong NP-completeness. It maps instances of one problem to instances of another in pseudo-polynomial
Pseudo-polynomial transformation
Pseudo-polynomial_transformation
input). UP contains P and is contained in NP. A common reformulation of NP states that a language is in NP if and only if a given "certificate" can be
UP_(complexity)
given allocation is CEEI is polynomial but checking if CEEI exists is co-NP-complete. Heinen et al extended the work of Bouveret and Lemaitre from additive
Fisher_market
Problem in graph theory
be NP-complete. It is easy to see that the problem is in NP: a yes answer is easy to prove by presenting a large enough cut. The NP-completeness of the
Maximum_cut
In contrast, deciding whether a given discrete allocation is PO is co-NP-complete. Therefore, if the divider claims that an allocation is fPO, the agents
Fractional_Pareto_efficiency
Extension of Datalog with integer arithmetic
Positive / Semi-positive: coNEXP-complete (combined complexity) and coNP-complete (data complexity). It captures \(\text{coNP}\) over ordered datasets
DatalogZ
Method for solving one problem using another
problem is NP-complete if it belongs to NP and all problems in NP have polynomial-time many-one reductions to it. A problem that belongs to NP can be proven
Polynomial-time_reduction
Subset of a graph's nodes such that all other nodes link to at least one
whether γ(G) ≤ K for a given graph G and input K; it is a classical NP-complete decision problem in computational complexity theory. Therefore it is
Dominating_set
problems are strongly NP-hard by reduction from 3-partition; When jobs can have one of two deadlines, the problems are NP-complete, by reduction from partition
Single-machine_scheduling
American computer scientist (born 1963)
language in co-NP that does not have an interactive protocol. In November 1989, Fortnow received an email from Noam Nisan showing that co-NP had multiple
Lance_Fortnow
Mathematical and computational problem
the problem is NP-hard, and the corresponding decision problem, deciding if items can fit into a specified number of bins, is NP-complete. Despite its worst-case
Bin_packing_problem
Subdivision of vertices into disjoint sets
partitioning or a balanced graph partition problem can be shown to be NP-complete to approximate within any finite factor. Even for special graph classes
Graph_partition
Algebraic structure used in logic
who showed it was PSPACE-complete and hence at least as hard as deciding equations of Boolean algebra (shown coNP-complete in 1971 by Stephen Cook) and
Heyting_algebra
Convex polytope whose vertices all have integer Cartesian coordinates
the complexity class coNP of problems for which a negative answer can be easily proven. More specifically, it is coNP-complete. Many of the important
Integral_polytope
Set of problems in computational complexity theory
Intractability: A Guide to the Theory of NP-Completeness. New York: W. H. Freeman & Co., 1979. The standard reference on NP-Complete problems - an important category
Complexity_class
Notation in general relativity
used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis
Newman–Penrose_formalism
Process of electing more than one winner in the same election / district
criteria, such as by their Borda count. It is coNP-complete to check if a committee satisfies this criterion, and coNP-hard to decide if there exists a Condorcet
Multiwinner_voting
Proof that a number is prime
evidence that these problems are not NP-complete, since if they were, it would imply that NP is subset of co-NP, a result widely believed to be false;
Primality_certificate
Complexity class
or even impossible to prove NP-hardness for TFNP problems. For example, if any TFNP problem is NP-complete, then NP = coNP, which is generally conjectured
TFNP
Economical computational problem
(given in the input)? This is NP-hard (NP-complete for 2 players). Is there a unique NE? This is NP-hard (CoNP-complete for 2 players). Is there a NE
Nash_equilibrium_computation
Intersection graph of unit disks in the plane
grows exponentially as a function of the dimension. It is NP-hard (more specifically, complete for the existential theory of the reals) to determine whether
Unit_disk_graph
Class of problems solvable in polynomial time
which this is true for the "no" instances is called co-NP. P is trivially a subset of NP and of co-NP; most experts believe it is a proper subset, although
P_(complexity)
Criterion of fair item allocation
whether a given allocation is MMS-fair is co-NP complete for agents with additive valuations (it is in co-NP, since it is possible to prove in polynomial
Maximin_share
List of unsolved computational problems
computational theory. What is the relationship between BQP and NP? NC = P problem NP = co-NP problem P = BPP problem P = PSPACE problem L = NL problem PH
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Type of computational problem
addition, if TFNP contains any FNP-complete problem it follows that N P = co-NP {\displaystyle \mathbf {NP} ={\textbf {co-NP}}} . Decision problem Search problem
Function_problem
polynomially bounded. The set of propositional tautologies, TAUT, is a coNP-complete set. A propositional proof system is a certificate-verifier for membership
Propositional_proof_system
American-Canadian computer scientist, contributor to complexity theory
NP-completeness, and proved the existence of an NP-complete problem by showing that the Boolean satisfiability problem (usually known as SAT) is NP-complete
Stephen_Cook
Class of computational complexity
PSPACE are the PSPACE-complete problems. See PSPACE-complete for examples of problems that are suspected to be in PSPACE but not in NP. The class PSPACE is
PSPACE
Data mining technique for simultaneous clustering of the rows and columns of a matrix
Bicluster. However, the most interesting variants of this problem are NP-complete. NP-complete has two conditions. In the simple case that there is an only element
Biclustering
Optimization problem
requiring a single route to visit all locations. As the TSP is NP-hard, the VRP is also NP-hard. VRP has many direct applications in industry. Vendors of
Vehicle_routing_problem
Concept in computer science
contained in NP and co-NP. There is even an oracle in which B P P = E X P N P {\displaystyle {\mathsf {BPP}}={\mathsf {EXP}}^{\mathsf {NP}}} (and hence
BPP_(complexity)
Fair division problem for discrete items
the mFS of an agent is coNP-complete. The problem of deciding whether an mFS allocation exists is in N P N P {\displaystyle NP^{NP}} , but its exact computational
Fair_item_allocation
Graph layout on multiple half-planes
thickness of a graph is NP-hard. This follows from the fact that finding Hamiltonian cycles in maximal planar graphs is NP-complete. In a maximal planar
Book_embedding
Concept in computational complexity theory
decision process is NEXPTIME-complete. Game complexity NP EXPTIME Juris Hartmanis, Neil Immerman, Vivian Sewelson. Sparse Sets in NP-P: EXPTIME versus NEXPTIME
NEXPTIME
Phrase which functions as a noun
A noun phrase – or NP or nominal (phrase) – is a phrase that usually has a noun or pronoun as its head, and has the same grammatical functions as a noun
Noun_phrase
Logical problem studied in computer science
software testing. Since Boolean satisfiability is already NP-complete, the SMT problem is typically NP-hard, and for many theories it is undecidable. Researchers
Satisfiability modulo theories
Satisfiability_modulo_theories
On collapse of the polynomial hierarchy if NP is in non-uniform polynomial time class
size circuits for SAT or for other NP-complete problems. A proof that such circuits do not exist would imply that P ≠ NP. As P/poly contains all problems
Karp–Lipton_theorem
Smallest dimension where a graph can be represented as an intersection graph of boxes
known. However, finding such a representation may be difficult: it is NP-complete to test whether the boxicity of a given graph is at most some given value
Boxicity
Set in graph theory
The problem of determining the paired domination number of a graph is NP-complete. Haynes, Teresa W.; Slater, Peter J. (1998). "Paired-domination in graphs"
Paired_dominating_set
Difficulty measures for computer science problems
defined as follows: BH1 is NP. BH2k is the class of languages which are the intersection of a language in BH2k-1 and a language in coNP. BH2k+1 is the class
Boolean_hierarchy
American mathematician
Engineering (1992) for major contributions to the theory and application of NP-completeness, constructing efficient combinatorial algorithms, and applying probabilistic
Richard_M._Karp
Search engine from Google
March 20th, it was confirmed that the roll out of the spam update was complete. In March 2024, an automated code agent operated by Google inadvertently
Google_Search
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