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

摘要

The first output-sensitive algorithm for the Maximal Clique Listing problem was given by Tsukiyama et al. (SIAM J Comput 6(3): 505-517, 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 et al. showed it is O(mn). Makino and Uno (in: Hagerup, Katajainen (eds) Algorithm theory: SWAT 2004. Lecture notes in computer science, Springer, Berlin, pp 260-272, 2004) sharpened the time delay to O(n(omega)) by generating all the children of a node in one single shot, which is 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 n(2) nodes in one single shot. According to the current upper bounds on square and rectangular fast matrix multiplication, with this the time delay improves from O(n(2.3728639)) to O(n(2.093362)), keeping a polynomial work space.