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Task of computing complete subgraphs
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete
Clique_problem
Adjacent subset of an undirected graph
given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although
Clique_(graph_theory)
Complete subgraph added to a random graph
clique problem is the algorithmic problem of distinguishing random graphs from graphs that have a planted clique. This is a variation of the clique problem;
Planted_clique
Problem of determining if a Boolean formula could be made true
reduction from 3-SAT to the other problem. An example of a problem where this method has been used is the clique problem: given a CNF formula consisting
Boolean satisfiability problem
Boolean_satisfiability_problem
Partition of a graph's nodes into cliques
In graph theory, a clique cover or partition into cliques of a given undirected graph is a collection of cliques that cover the whole graph. Generally
Clique_cover
Unrelated vertices in graphs
NP-completeness to problems related to independent sets. The independent set problem and the clique problem are complementary: a clique in G is an independent
Independent set (graph theory)
Independent_set_(graph_theory)
all directed edges. Variants include the rural postman problem. Clique cover problem Clique problem Complete coloring, a.k.a. achromatic number Cycle rank
List_of_NP-complete_problems
Problem in theoretical computer science
Subgraph isomorphism is a generalization of both the maximum clique problem and the problem of testing whether a graph contains a Hamiltonian cycle, and
Subgraph_isomorphism_problem
Graph with tight clique-coloring relation
relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent
Perfect_graph
Area of discrete mathematics
certain kind is often an NP-complete problem. For example: Finding the largest complete subgraph is called the clique problem (NP-complete). One special case
Graph_theory
Algorithm for listing maximal cliques
its description in 1973. Although other algorithms for solving the clique problem have running times that are, in theory, better on inputs that have few
Bron–Kerbosch_algorithm
Historical warlord clique in China
The New Guangxi clique (Chinese: 新桂系; pinyin: Xīn Guìxì), led by Li Zongren, Huang Shaohong, and Bai Chongxi, was a warlord clique during the Republic
New_Guangxi_clique
Complexity class
Knapsack problem Hamiltonian path problem Travelling salesman problem (decision version) Subgraph isomorphism problem Subset sum problem Clique problem Vertex
NP-completeness
Estimate of time taken for running an algorithm
large clique in the union of a clique and a random graph. Although quasi-polynomially solvable, it has been conjectured that the planted clique problem has
Time_complexity
Chinese faction during the Warlord Era
The Anhui clique (Chinese: 皖系; pinyin: Wǎn Xì) was a military and political organization, one of several mutually hostile cliques or factions that split
Anhui_clique
Unsolved problem in computational complexity theory
A clique problem for a class of so-called M-graphs. It is shown that finding an isomorphism for n-vertex graphs is equivalent to finding an n-clique in
Graph_isomorphism_problem
induced tree The Hadwiger conjecture relating coloring to clique minors The Hadwiger–Nelson problem on the chromatic number of unit distance graphs Jaeger's
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Topics referred to by the same term
up clique in Wiktionary, the free dictionary. A clique is a close social group. Clique or The Clique may also refer to: Clique (graph theory) Clique problem
Clique_(disambiguation)
Subfield of mathematical optimization
Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the TSP and clique problem. An NPO problem is called polynomially bounded
Combinatorial_optimization
Solving an optimization problem with a quadratic objective function
maximum that is function of the clique number of G. Computing the clique number of a graph is a well-known NP-hard problem; hence, solving the quadratic
Quadratic_programming
20th-century Chinese army
Qing dynasty, and, by dividing into warlord factions known as the Beiyang clique (Chinese: 北洋軍閥; pinyin: Běiyáng Jūnfá), ushered in a period of regional
Beiyang_Army
Branch of computational complexity theory
The clique problem can be coded as Φ clique ( V , E ) := ⋀ { u , v } ⊂ V , u ≠ v , { u , v } ∉ E ( ¬ x u ∨ ¬ x v ) {\displaystyle \Phi _{\text{clique}}(V
Parameterized_complexity
Model of computational complexity
bits modulo some odd prime p. The k-clique problem is to decide whether a given graph on n vertices has a clique of size k. For any particular choice
Circuit_complexity
Independent set which is not a subset of any other independent set
A graph may have many maximal cliques, of varying sizes; finding the largest of these is the maximum clique problem. Some authors include maximality
Maximal_independent_set
Subset of a graph's vertices, including at least one endpoint of every edge
reduction from 3-satisfiability or, as Karp did, by reduction from the clique problem. Vertex cover remains NP-complete even in cubic graphs and even in planar
Vertex_cover
early days: compiler design problem, clique problem, very large scale integration (VLSI) design. Additionally, pricing problems appear to be an application
Quadratic_knapsack_problem
Set of computational problems stated by Richard Karp (1973)
restrictions must be satisfied, with no optimization) Clique (see also independent set problem) Set packing Vertex cover Set covering Feedback node set
Karp's 21 NP-complete problems
Karp's_21_NP-complete_problems
2012 single by Kanye West, Jay-Z and Big Sean
"Clique" is a song by American rappers Kanye West, Jay-Z, and Big Sean from West's record label GOOD Music's debut compilation album, Cruel Summer (2012)
Clique_(song)
Graph made from a subset of another graph's nodes and their edges
be found as an induced subgraph of another. Because it includes the clique problem as a special case, it is NP-complete. Diestel, Reinhard (2006), Graph
Induced_subgraph
Logical formulation of graph properties
the clique problem (for a fixed clique size) may be expressed by a sentence that states the existence of a number of vertices equal to the clique size
Logic_of_graphs
Fewest cliques covering a graph's edges
called the intersection number problem, the intersection graph basis problem, covering by cliques, the edge clique cover problem, and (because of one of its
Intersection number (graph theory)
Intersection_number_(graph_theory)
List of unsolved computational problems
NP? NC = P problem NP = co-NP problem P = BPP problem P = PSPACE problem L = NL problem PH = PSPACE problem L = P problem L = RL problem Unique games
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Methodic assignment of colors to elements of a graph
\omega (G).} For perfect graphs this bound is tight. Finding cliques is known as the clique problem. Hoffman's bound: Let W {\displaystyle W} be a real symmetric
Graph_coloring
Cliques that develop among adolescents
Adolescent cliques are cliques that develop amongst adolescents. In the social sciences, the word "clique" is used to describe a large group of 6 to 12
Adolescent_clique
The MaxCliqueDyn algorithm is an algorithm for finding a maximum clique in an undirected graph. MaxCliqueDyn is based on the MaxClique algorithm, which
MaxCliqueDyn_algorithm
Geometry problem on tiling by hypercubes
dimensions. The proofs of these results use a reformulation of the problem in terms of the clique number of certain graphs now known as Keller graphs. The related
Keller's_conjecture
Problem in computer science
version (general maximum set packing problem) has been proven as difficult to approximate as the maximum clique problem; in particular, it cannot be approximated
Set_packing
Largest independent set of paired elements
NP-complete. This problem transformation does not depend on the structure of the clique problem in any deep way, and would work for any other problem of finding
Matroid_parity_problem
Unproven computational hardness assumption
algorithm for edge clique cover must have double exponential dependence on the parameter. In the three-party set disjointness problem in communication complexity
Exponential_time_hypothesis
Computer science problem
longest increasing subsequence algorithms can be used to solve the clique problem efficiently in permutation graphs. In the Robinson–Schensted correspondence
Longest increasing subsequence
Longest_increasing_subsequence
Chinese warlord faction in control of Manchuria (1911–1928)
remnants of the Anhui clique, which had become allies of the Fengtian Clique, which resulted in a Fengtian victory, with the Zhili clique retreating as far
Fengtian_clique
maximal cliques rather than the individual k-cliques. This inevitably requires finding the graph's maximum clique, which is an NP-hard problem. (We emphasize
Clique_percolation_method
traveller problem Cliques and independent sets Clique problem Connected component Cycle space de Bruijn sequences Degree diameter problem Entanglement
List_of_graph_theory_topics
Computational complexity class
time hypothesis. Other problems for which the best known algorithm takes quasi-polynomial time include: The planted clique problem, of determining whether
Quasi-polynomial_time
Political faction in China
The Shanghai clique, also referred to as the Shanghai gang, Jiang clique, or Jiang faction, refers to an informal group of Chinese Communist Party (CCP)
Shanghai_clique
Complexity class
optimization problem Minimum vertex cover Maximum clique Longest simple path Graph coloring; an application: register allocation in compilers Lists of problems Lists
NP-hardness
neighbourhood of each vertex is a comparability graph implies that the Maximum Clique problem is polynomially solvable on word-representable graphs. Semi-transitive
Word-representable_graph
NP-complete graph problem
larger graph, and the maximum clique problem is an induced subgraph isomorphism problem in which one seeks to find a large clique graph as an induced subgraph
Induced subgraph isomorphism problem
Induced_subgraph_isomorphism_problem
Hypothesis in computational complexity theory
like the planted clique conjecture. However, for cryptographic applications, knowing that a problem has some hard instance (the problem is hard in the worst-case)
Computational hardness assumption
Computational_hardness_assumption
Dutch computer scientist (1937–2006)
with Joep Kerbosch he invented the Bron–Kerbosch algorithm for the clique problem. Born in Amsterdam, Bron read Chemistry at Utrecht University. After
Coenraad_Bron
American rapper
2025-12-21. Barlas, Jon (2025-03-06). "Thirteendegrees shares 'DA PROBLEM SOLVA' visual off 'Clique City Vol. 2'". Our Generation Music. Retrieved 2025-12-21
Thirteendegrees
Cantor–Bernstein–Schroeder theorem Cayley's formula Cayley's theorem Clique problem (to do) Compactness theorem (very compact proof) Erdős–Ko–Rado theorem
List_of_mathematical_proofs
Overview of and topical guide to algorithms
algorithm Graph coloring Clique problem Independent set (graph theory) Hamiltonian path problem Travelling salesman problem String-searching algorithm
Outline_of_algorithms
Highly connected subgraph
the maximum density subgraph on exactly k vertices. This problem generalizes the clique problem and is thus NP-hard in general graphs. There exists a polynomial
Dense_subgraph
NP-hard problems, including the travelling salesman problem, graph coloring, maximum clique problem, and minimal vertex cover. These problems are only
Disjunctive_Datalog
Graph representing a permutation
decreasing subsequence in the permutation defining the graph, so the clique problem may be solved in polynomial time for permutation graphs by using a longest
Permutation_graph
Conjecture in graph theory
Unsolved problem in mathematics Do the graphs with a fixed forbidden induced subgraph necessarily have large cliques or large independent sets? More unsolved
Erdős–Hajnal_conjecture
Class of algorithms that find approximate solutions to optimization problems
even polynomial, factor unless P = NP, as in the case of the maximum clique problem. Therefore, an important benefit of studying approximation algorithms
Approximation_algorithm
Graph where all long cycles have a chord
elimination orderings is finding a maximum clique of a chordal graph in polynomial-time, while the same problem for general graphs is NP-complete. More generally
Chordal_graph
Unsolved problem in discrete geometry
The big-line-big-clique conjecture is an unsolved problem in discrete geometry, stating that finite sets of many points in the Euclidean plane either have
Big-line-big-clique conjecture
Big-line-big-clique_conjecture
Problem of finding the longest simple path for a given graph
depends on the clique-width of the graph, so this algorithms is not fixed-parameter tractable. The longest path problem, parameterized by clique-width, is
Longest_path_problem
Gluing graphs at complete subgraphs
a branch of mathematics, a clique sum (or clique-sum) is a way of combining two graphs by gluing them together at a clique, analogous to the connected
Clique-sum
Measure of graph complexity
approximation algorithms for clique-width are known. Based on these algorithms and on Courcelle's theorem, many graph optimization problems that are NP-hard for
Clique-width
graphs, many algorithmic problems that are hard on other classes of graphs, including graph coloring and the clique problem, can be solved in polynomial
Tolerance_graph
hierarchy Clique problem Hamiltonian cycle problem Hamiltonian path problem Integer factorization Knapsack problem Satisfiability problem 2-satisfiability
List of computability and complexity topics
List_of_computability_and_complexity_topics
problem is a tree decomposition of its primal graph. Every constraint can be enforced on a node of the tree because each constraint creates a clique on
Decomposition method (constraint satisfaction)
Decomposition_method_(constraint_satisfaction)
American dance music group
Heart", which was released in 1990 from their debut studio album, World Clique (1990), was a top-ten hit in multiple countries. In December 2016, Billboard
Deee-Lite
Problem in graph theory
certain classes of graphs, the algorithms for this problem can be extended to the 2- and 3-clique-sums of graphs in these classes. This allows the planar
Maximum_cut
Danish association football club
attention", as well as cliques inside the first team. In order to solve the clique problems, he had gone to director Per Bjerregaard to fire three key players –
Brøndby_IF
have few cliques if every member of the class has a polynomial number of maximal cliques. Certain generally NP-hard computational problems are solvable
Graphs_with_few_cliques
Period in the history of the Republic of China (1916–1928)
The most powerful cliques were the Zhili clique led by Feng Guozhang, who controlled several northern provinces; the Anhui clique led by Duan Qirui,
Warlord_Era
of envious agents. The proof is by reduction from Maximum clique problem. However, the problem is fixed-parameter tractable when parameterized by the total
House_allocation_problem
Complexity of sending information in a distributed algorithm
algorithms based on linear programming that approximately solve the maximum clique problem. Omri Weinstein's 2015 survey surveys the subject. Quantum communication
Communication_complexity
2012 single by ASAP Rocky featuring 2 Chainz, Drake and Kendrick Lamar
rapper Tyga has a freestyle to the song on his 2012 mixtape, 187, alongside "Clique" by Kanye West. Lil Wayne freestyled over the beat on his Dedication 5 mixtape
Fuckin'_Problems
Intersection graph of trapezoids between parallel lines
independent set problem and an O ( n 2 log n ) {\displaystyle {O}(n^{2}\log n)} algorithm for the maximum weighted clique problem. k-Trapezoid graphs
Trapezoid_graph
defined by an excluded induced subgraph, every graph has either a large clique or a large independent set. The Erdős–Mollin–Walsh conjecture on consecutive
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
1988; 391–401. E. Balas, V. Chvátal, J. Nesetril: On The Maximum-Weight Clique Problem, Mathematics of Operations Research 12, 1987; 522–536. E. Balas: Disjunctive
Egon_Balas
Smallest complete lattice containing a partial order
or a maximal clique in the complement of the comparability graph, so algorithms for the clique problem or the independent set problem can also be applied
Dedekind–MacNeille_completion
One possible solution for this problem is to build a modular product graph of G and H. In this graph, the largest clique corresponds to a maximum common
Maximum common induced subgraph
Maximum_common_induced_subgraph
Series of young adult novels by Lisi Harrison
The Clique is a young adult novel series written by Canadian author Lisi Harrison and originally published by Little, Brown and Company, a subsidiary of
The_Clique_(series)
Unsolved problem in extremal graph theory
subgraphs of a given size? More unsolved problems in mathematics The Zarankiewicz problem, an unsolved problem in mathematics, asks for the largest possible
Zarankiewicz_problem
Intersection graph of unit disks in the plane
efficiently by using the geometric structure of these graphs, and the maximum clique problem can be solved exactly for these graphs in polynomial time, given a disk
Unit_disk_graph
Conjecture about coloring graphs
belong to a single clique, as their coloring presents no difficulty; once this is done, the hypergraph that has a vertex for each clique, and a hyperedge
Erdős–Faber–Lovász_conjecture
Government of the early Republic of China (1912–1928)
became known as the Anhui clique. His detractors rallied around President Feng and formed the Zhili clique. The Zhili clique favored peaceful negotiations
Beiyang_government
{\displaystyle H} . The problem also generalizes several other well-known graph problems, including maximum clique and maximum path. The problem is APX-hard, unless
Maximum_common_edge_subgraph
Computer science string term
words plays a key role in the proof of hardness of approximation of the clique problem, in which a collection of partial words representing successful runs
Partial_word
Subdivision of vertices into disjoint sets
Recently, the graph partition problem has gained importance due to its application for clustering and detection of cliques in social, pathological and biological
Graph_partition
Kayles game and clique-forming game: two players alternately select vertices and the induced subgraph must be an independent set (resp. clique). The last to
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
Independent set in a graph
each department. Thus this problem is sometimes called the happy dean problem. A rainbow clique or a colorful clique is a clique in which every vertex has
Rainbow-independent_set
Graph representing structure of another graph's cliques
In graph theory, a clique graph of an undirected graph G is another graph K(G) that represents the structure of cliques in G. Clique graphs were discussed
Clique_graph
Set of random variables
\operatorname {cl} (G)} is the set of cliques of G {\displaystyle G} . The definition is equivalent if only maximal cliques are used. The functions φ C {\displaystyle
Markov_random_field
Type of computational problem
in which GO is made of disjoint cliques, where each clique represents a color. Conflict-free set cover is the problem of finding a conflict-free subset
Covering_problems
Size of largest complete graph made by contracting edges of a given graph
number is NP-hard. However, the problem is fixed-parameter tractable: there is an algorithm for finding the largest clique minor in an amount of time that
Hadwiger_number
Mapping of a graph into a tree
computational problems on the graph. Tree decompositions are also called junction trees, clique trees, or join trees. They play an important role in problems like
Tree_decomposition
Graph which partitions into a clique and independent set
three different ways: the clique {a, b} and the independent set {c} the clique {b, c} and the independent set {a} the clique {b} and the independent set
Split_graph
Type of computational problem
a counting problem is a type of computational problem that is obtained by strengthening a decision problem. For example, the SAT problem asks: "Given
Counting_problem_(complexity)
Concept in graph theory
known as a "Clique graph". The clique graphs have vertices which represent the cliques in the original graph while the edges of the clique graph record
Community_structure
Type of algorithm
optimization variant of the k-Clique problem is the Densest k-Subgraph problem (which is a 2-ary Constraint Satisfaction problem), where the task is to find
Parameterized approximation algorithm
Parameterized_approximation_algorithm
Warlord factions in Guizhou
The Guizhou clique, also known as the Qian clique (Qian being the abbreviated name of Guizhou; Chinese: 黔系; pinyin: Qiánxì; Wade–Giles: Chʻien2-hsi4)
Guizhou_in_the_Warlord_Era
American musician
and witch house. He has worked as a member of collectives such as GothBoiClique, Thraxxhouse and Misery Club. He also played in the indie rock band Pay
Wicca_Phase_Springs_Eternal
CLIQUE PROBLEM
CLIQUE PROBLEM
Girl/Female
Tamil
Unique
Boy/Male
Indian
Sweet; Flower; Kind to All
Male
English
English surname transferred to forename use, from the name of various places, derived from Old English clif, CLIVE means "bank, cliff, slope."
Girl/Female
Tamil
Annjaya | அநà¯à®¨à¯à®œà®¯
Unique
Annjaya | அநà¯à®¨à¯à®œà®¯
Male
Spanish
 Pet form of Spanish Enrique, QUIQUE means "home-ruler." Compare with another form of Quique.
Boy/Male
Hindu, Indian
Messenger of God
Girl/Female
Latin American
Only one.
Girl/Female
Hindu, Indian, Unique
Goddess Lakshmi; Requester; Unique
Surname or Lastname
English
English : variant spelling of Cliff.
Surname or Lastname
English
English : habitational name from any of various places, for example in Shropshire and Cheshire, named Clive, from the dative case of Old English clif ‘slope’, ‘bank’, ‘cliff’ (see Cliff), originally used after a preposition. In some cases the name may be topographical, with the same origin and meaning.
Boy/Male
French
Girl/Female
Tamil
Unique
Boy/Male
Shakespearean
The Tragedy of Julius Caesar' A servant to Brutus.
Boy/Male
American, Australian, British, Chinese, Christian, Danish, English, French, German, Jamaican
Cliff; Slope; Bank; Someone who Lived Near a Cliff or a Riverbank; Form of Clifford; From a Steep Cliff
Girl/Female
Tamil
Unique
Male
English
Cliff Dweller
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Unique Oneness; Unique
Girl/Female
Tamil
Unique
Female
French
Feminine form of French Frédéric, FRÉDÉRIQUE means "peaceful ruler."
Boy/Male
English
Cliff. Lives at the cliffs. From an English surname and place name. Famous bearer 18th century...
CLIQUE PROBLEM
CLIQUE PROBLEM
Girl/Female
Christian & English(British/American/Australian)
Powerful, Regal
Boy/Male
Hindu
Time
Boy/Male
Indian
Super Strong
Girl/Female
Indian
Full Moon
Boy/Male
Assamese, Hindu, Indian, Kannada, Marathi, Sanskrit, Telugu
One who is in Control of Destiny
Surname or Lastname
English
English : variant of Shapley.Thomas Shapleigh (1765–1800), born in Kittery MA, was librarian of Harvard College in the 1790s.
Girl/Female
Hindu
Another name of Parvati
Male
Norse
 Old Norse name composed of the name of the god Thor and the word ketill "cauldron," hence "Þórr's cauldron."
Girl/Female
Latin
Of Mars. Mars was mythological Roman god of fertility for whom the month March was named;...
Surname or Lastname
English and northern Irish
English and northern Irish : possibly from a diminutive of the nickname totte ‘simpleton’, ‘fool’.
CLIQUE PROBLEM
CLIQUE PROBLEM
CLIQUE PROBLEM
CLIQUE PROBLEM
CLIQUE PROBLEM
v. i.
A narrow circle of persons associated by common interests or for the accomplishment of a common purpose; -- generally used in a bad sense.
a.
Oblique; sloping.
n.
A light skiff or rowboat used on the Bosporus; also, a Levantine vessel of larger size.
v. i.
To deviate from a perpendicular line; to move in an oblique direction.
v. t.
To excite to action by causing resentment or jealousy; to stimulate; to prick; as, to pique ambition, or curiosity.
p. pr. & vb. n.
of Oblique
n.
See Cazique.
n.
A genus of naked pteropods. One species (Clione papilonacea), abundant in the Arctic Ocean, constitutes a part of the food of the Greenland whale. It is sometimes incorrectly called Clio.
n.
An assembly or clique of pedants.
n.
An oblique line.
a.
Having oblique angles; as, an oblique-angled triangle.
n.
See Cinque.
a.
Of or pertaining to a clique; disposed to from cliques; exclusive in spirit.
v. i.
To To associate together in a clannish way; to act with others secretly to gain a desired end; to plot; -- used with together.
n.
The tendency to associate in cliques; the spirit of cliques.
imp. & p. p.
of Oblique
n.
See Caique.
n.
Same as Silique.
v. i.
To march in a direction oblique to the line of the column or platoon; -- formerly accomplished by oblique steps, now by direct steps, the men half-facing either to the right or left.
n.
A collection of persons employed to applaud at a theatrical exhibition.