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

• Published in: Algorithmica Vol. 80; no. 12; pp. 3525 - 3562
• Main Authors: Comin, Carlo, Rizzi, Romeo
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2018; Springer Nature B.V; Springer Verlag
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Children; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Algorithms; Data Structures and Information Theory; Delay; Mathematics of Computing; Matrix; Multiplication; Multiplication & division; Nodes; Theory of Computation; Time lag; Upper bounds; Rectangular fast matrix multiplication; Polynomial time delay; Reverse search enumeration; Maximal clique listing; Output sensitive
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-017-0402-5

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 ω ) 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.