The brute force algorithm finds a 4-clique in this 7-vertex graph (the complement of the 7-vertex path graph) by systematically checking all C(7,4) = 35 4-vertex subgraphs for completeness.
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) 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 (a clique with the largest possible number of vertices), finding a maximum weight clique in a weighted graph, listing all maximal cliques (cliques that cannot be enlarged), and solving the decision problem of testing whether a graph contains a clique larger than a given size.
The clique problem arises in the following real-world setting. Consider a social network, where the graph's vertices represent people, and the graph's edges represent mutual acquaintance. Then a clique represents a subset of people who all know each other, and algorithms for finding cliques can be used to discover these groups of mutual friends. Along with its applications in social networks, the clique problem also has many applications in bioinformatics, and computational chemistry.
Most versions of the 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 both fixed-parameter intractable and hard to approximate. And, listing all maximal cliques may require exponential time as there exist graphs with exponentially many maximal cliques. Therefore, much of the theory about the clique problem is devoted to identifying special types of graph that admit more efficient algorithms, or to establishing the computational difficulty of the general problem in various models of computation.
To find a maximum clique, one can systematically inspect all subsets, but this sort of brute-force search is too time-consuming to be practical for networks comprising more than a few dozen vertices.
Although no polynomial time algorithm is known for this problem, more efficient algorithms than the brute-force search are known. For instance, the Bron–Kerbosch algorithm can be used to list all maximal cliques in worst-case optimal time, and it is also possible to list them in polynomial time per clique.
In computer science, the cliqueproblem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete...
Subgraph isomorphism is a generalization of both the maximum cliqueproblem and the problem of testing whether a graph contains a Hamiltonian cycle, and...
reduction from 3-SAT to the other problem. An example of a problem where this method has been used is the cliqueproblem: given a CNF formula consisting...
cliqueproblem is the algorithmic problem of distinguishing random graphs from graphs that have a planted clique. This is a variation of the clique problem;...
certain kind is often an NP-complete problem. For example: Finding the largest complete subgraph is called the cliqueproblem (NP-complete). One special case...
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...
relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum cliqueproblem, and maximum independent...
complement of G. The clique cover problem in computational complexity theory is the algorithmic problem of finding a minimum clique cover, or (rephrased...
Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the TSP and cliqueproblem. An NPO problem is called polynomially bounded...
The MaxCliqueDyn algorithm is an algorithm for finding a maximum clique in an undirected graph. MaxCliqueDyn is based on the MaxClique algorithm, which...
Adolescent cliques are cliques that develop amongst adolescents. In the social sciences, the word "clique" is used to describe a group of 3 to 12 "who...
large clique in the union of a clique and a random graph. Although quasi-polynomially solvable, it has been conjectured that the planted cliqueproblem has...
equivalent to finding an n-clique in an M-graph of size n2. This fact is interesting because the problem of finding a clique of order (1 − ε)n in a M-graph...
be found as an induced subgraph of another. Because it includes the cliqueproblem as a special case, it is NP-complete. Diestel, Reinhard (2006), Graph...
bits modulo some odd prime p. The k-cliqueproblem is to decide whether a given graph on n vertices has a clique of size k. For any particular choice...
\omega (G).} For perfect graphs this bound is tight. Finding cliques is known as the cliqueproblem. Hoffman's bound: Let W {\displaystyle W} be a real symmetric...
reduction from 3-satisfiability or, as Karp did, by reduction from the cliqueproblem. Vertex cover remains NP-complete even in cubic graphs and even in planar...
The Fengtian clique (Chinese: 奉系军阀; pinyin: Fèngxì Jūnfá; Wade–Giles: Feng-hsi Chün-fa) was the faction that supported warlord Zhang Zuolin during China's...
version (general maximum set packing problem) has been proven as difficult to approximate as the maximum cliqueproblem; in particular, it cannot be approximated...
Generalized star-height problem Separating words problem Fellows, Michael R.; Rosamond, Frances A.; Rotics, Udi; Szeider, Stefan (2009), "Clique-width is NP-complete"...
traveller problemCliques and independent sets Cliqueproblem Connected component Cycle space de Bruijn sequences Degree diameter problem Entanglement...
longest increasing subsequence algorithms can be used to solve the cliqueproblem efficiently in permutation graphs. In the Robinson–Schensted correspondence...
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