Improved lower bounds on the randomized complexity of graph properties

• Published in: Random structures & algorithms Vol. 30; no. 3; pp. 427 - 440
• Main Authors: Chakrabarti, Amit, Khot, Subhash
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.05.2007
• Subjects: complexity; decision trees; graph packing; graph properties; probabilistic method; randomized algorithms
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.20164

We prove a lower bound of Ω(n4/3 log 1/3n) on the randomized decision tree complexity of any nontrivial monotone n‐vertex graph property, and of any nontrivial monotone bipartite graph property with bipartitions of size n. This improves the previous best bound of Ω(n4/3) due to Hajnal (Combinatorica 11 (1991) 131–143). Our proof works by improving a graph packing lemma used in earlier work, and this improvement in turn stems from a novel probabilistic analysis. Graph packing being a well‐studied subject in its own right, our improved packing lemma and the probabilistic technique used to prove it may be of independent interest. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007