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

International audienceThe 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

Listing Maximal Independent Sets with Minimal Space and Bounded Delay

International audienceAn independent set is a set of nodes in a graph such that no two of them are adjacent. It is maximal if there is no node outside the independent set that may join it. Listing maximal independent sets in graphs can be applied, for example, to sample nodes belonging to different communities or clusters in network analysis and document clustering. The problem has a rich history as it is related to maximal cliques, dominance sets, vertex covers and 3-colorings in graphs. We are interested in reducing the delay, which is the worst-case time between any two consecutively output solutions, and the memory footprint, which is the additional working space behind the read-only input graph