A note on the random greedy independent set algorithm

• Published in: Random structures & algorithms Vol. 49; no. 3; pp. 479 - 502
• Main Authors: Bennett, Patrick, Bohman, Tom
• Format: Journal Article
• Language: English
• Published: Hoboken Blackwell Publishing Ltd 01.10.2016; Wiley Subscription Services, Inc
• Subjects: Additives; Algorithms; Collection; Combinatorial analysis; Constants; Graph theory; Greedy algorithms; hypergraph independence number; Lower bounds; randomized algorithms; Turan problem
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.20667

Let r be a fixed constant and let H be an r‐uniform, D‐regular hypergraph on N vertices. Assume further that D>Nϵ for some ϵ>0. Consider the random greedy algorithm for forming an independent set in H. An independent set is chosen at random by iteratively choosing vertices at random to be in the independent set. At each step we chose a vertex uniformly at random from the collection of vertices that could be added to the independent set (i.e. the collection of vertices v with the property that v is not in the current independent set I and I∪{v} contains no edge in H). Note that this process terminates at a maximal subset of vertices with the property that this set contains no edge of H; that is, the process terminates at a maximal independent set. We prove that if H satisfies certain degree and codegree conditions then there are Ω(N·((logN)/D)1r−1) vertices in the independent set produced by the random greedy algorithm with high probability. This result generalizes a lower bound on the number of steps in the H‐free process due to Bohman and Keevash and produces objects of interest in additive combinatorics. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 479–502, 2016