The.Algorithm.Design.Manual.Springer-Verlag.1998

Clique
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Clique
Input description: A graph G=(V,E).
Problem description: What is the largest such that for all , ?
Discussion: When I went to high school, everybody complained about the ``clique'', a group of friends
who all hung around together and seemed to dominate everything social. Consider a graph whose
vertices represent a set of people, with edges between any pair of people who are friends. Thus the clique
in high school was in fact a clique in this friendship graph.
Identifying ``clusters'' of related objects often reduces to finding large cliques in graphs. One example is
in a program recently developed by the Internal Revenue Service (IRS) to detect organized tax fraud,
where groups of phony tax returns are submitted in the hopes of getting undeserved refunds. The IRS
constructs graphs with vertices corresponding to submitted tax forms and with edges between any two
tax-forms that appear suspiciously similar. A large clique in this graph points to fraud.
Since any edge in a graph represents a clique of two vertices, the challenge lies not in finding a clique,
but in finding a large one. And it is indeed a challenge, for finding a maximum clique is NP-complete.
To make matters worse, not only is no good approximation algorithm known, it is provably hard to
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