An Improved Upper Bound on Maximal Clique Listing via Rectangular Fast Matrix Multiplication

Abstract

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 $O(mn)$. In 2004, Makino and Uno sharpened the time delay to $O(n^{\omega})$ by generating all the children of a node in one single shot performed by computing a \emph{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 $n^2$ nodes in one shot. According to the current upper bounds on fast matrix multiplication, with this the time delay improves from $O(n^{2.3728639})$ to $O(n^{2.093362})$.