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Finding Maximum k-Cliques Faster Using Lazy Global Domination

Last modified: 2016-06-20

#### Abstract

A clique in a graph is a set of vertices, each of which is adjacent to every other vertex in this set. A

*k*-clique relaxes this requirement, requiring vertices to be within a distance*k*of each other, rather than directly adjacent. In theory, a maximum clique algorithm can easily be adapted to solve the maximum*k*-clique problem, although large sparse*k*-clique graphs reduce to large dense clique graphs, which can be computationally challenging. We adapt a state of the art maximum clique algorithm to show that this reduction is in fact useful in practice, and introduce a lazy global domination rule which sometimes vastly reduces the search space. We include experimental results for a range of real-world and benchmark graphs, and a detailed look at random graphs. We also use thread-parallel search to solve some harder instances.
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