The first output-sensitive algorithm for the Maximal Clique Listing problem was given by Tsukiyama et. al. in 1977. As any algorithm falling within the Reverse Search paradigm, it performs a DFS visit of a directed tree (the RS-tree) having the objects to be listed (i.e. maximal cliques) as its nodes. In a recursive implementation, the RS-tree corresponds to the recursion tree of the algorithm. The time delay is given by the cost of generating the next child of a node, and Tsukiyama showed it is . In 2004, Makino and Uno sharpened the time delay to by generating all the children of a node in one single shot performed by computing a *square* fast matrix multiplication. In this paper, we further improve the asymptotics for the exploration of the same RS-tree by grouping the offsprings’ computation even further. Our idea is to rely on *rectangular* fast matrix multiplication in order to compute all children of nodes in one shot. According to the current upper bounds on square and rectangular fast matrix multiplication, with this the time delay improves from to .

Download: Manuscript.

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Reblogged this on Carlo 's Blog and commented:

improved version