Regularity Lemmas and Combinatorial Algorithms

— We present new combinatorial algorithms for Boolean matrix multiplication (BMM) and preprocessing a graph to answer independent set queries. We give the first asymptotic improvements on combinatorial algorithms for dense BMM in many years, improving on the “Four Russians” O(n3 /(w log n)) bound for machine models with wordsize w. (For a pointer machine, we can set w = log n.) The algorithms utilize notions from Regularity Lemmas for graphs in a novel way. • We give two randomized combinatorial algorithms for BMM. The first algorithm is essentially a reduction from BMM to the Triangle Removal Lemma. The best known bounds for the Triangle Removal Lemma only imply an O ` (n3 log β)/(βw log n) ´ time algorithm for BMM where β = (log∗ n)δ for some δ > 0, but improvements on the Triangle Removal Lemma would yield corresponding runtime improvements. The second algorithm applies the Weak Regularity Lemma of Frieze and Kannan along with several information compression ide...

• Type: Conference
• Year: 2009
• Where: FOCS
• Authors: Nikhil Bansal, Ryan Williams
• Keywords: Combinatorial Algorithms, FOCS 2009, Regularity Lemma, Theoretical Computer Science, Triangle Removal Lemma