Finding a Heaviest Triangle is not Harder than Matrix Multiplication

We show that for any > 0, a maximum-weight triangle in an undirected graph with n vertices and real weights assigned to vertices can be found in time O(n + n2+), where is the exponent of fastest matrix multiplication algorithm. By the currently best bound on , the running time of our algorithm is O(n2.376). Our algorithm substantially improves the previous time-bounds for this problem recently established by Vassilevska et al. (STOC 2006, O(n2.688)) and (ICALP 2006, O(n2.575)). Its asymptotic time complexity matches that of the fastest known algorithm for finding a triangle (not necessarily a maximum-weight one) in a graph. By applying or extending our algorithm, we can also improve the upper bounds on finding a maximum-weight triangle in a sparse graph and on finding a maximum-weight subgraph isomorphic to a fixed graph established in the papers by Vassilevska et al. For example, we can find a maximum-weight triangle in a vertex-weighted graph with m edges in asymptotic time requ...

• Type: Journal
• Year: 2006
• Where: ECCC
• Authors: Artur Czumaj, Andrzej Lingas
• Keywords: Algorithm, ECCC 2006, Matrix Multiplication Algorithm, Maximum-weight Triangle