Breaking the hegemony of the triangle method in clique detection
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We consider the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a straightforward reduction to triangle detection/counting. We provide an alternative method of detection/counting copies of fixed size cliques based on a multi-dimensional matrix product. It is at least as time efficient as the triangle method in cases of K_4 and K_5. The complexity of the multi-dimensional matrix product is of interest in its own rights. We provide also another alternative method for detection/counting K_r copies, again time efficient for r∈{ 4, 5 }.
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