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Let's say you have a graph like this:

example

The largest subset of nodes so that every pair of vertices have an edge between them is $\{3, 5, 7, 12\}$. The second largest subsets are

  • $\{4, 7, 21\}$,
  • $\{3, 5, 7\}$,
  • $\{3, 5, 12\}$,

etc.

Is there an algorithm to find the largest subset efficiently?

Also, keep in mind that I know virtually nothing about graphs (haven't learned at school, haven't taught myself). The data was originally given as a set of pairs of numbers.

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    $\begingroup$ This problem is called Maximum Clique, and unfortunately it's NP-hard, meaning it's very unlikely that any algorithm exists that can solve every instance quickly. $\endgroup$ – j_random_hacker Jun 2 '18 at 14:49
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    $\begingroup$ If you want to find a maximum clique (or even enumerate all of them), there are good solvers for that available. $\endgroup$ – Juho Jun 3 '18 at 9:55
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Unfortunately, the answer is no.

In graph theory, such a subset is called a clique. (Essentially, a graph is a set of nodes and a set of edges between those nodes, and a clique is a set of nodes in a graph such that every node in the set has an edge to every other node in the set.) What you want is to find the largest clique in a given graph.

This is called the Maximum Clique Problem, and it's NP-hard. The decision problem version, "is there a clique with size $k$?", is in fact NP-complete. So an efficient solution to the Maximum Clique Problem would prove that P=NP and earn you the $1,000,000 prize and eternal fame.

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