We study the problem of finding a maximal independent set (MIS) in the standard LOCAL model of distributed computing. Classical algorithms by Luby JACM'86 and Alon, Babai, and Itai JALG'86 find an MIS in O (n) rounds in n-node graphs with high probability. Despite decades of research, the existence of any o (n) -round algorithm for general graphs remains one of the major open problems in the field. Interestingly, the hard instances for this problem must contain constant-length cycles. This is because there exists a sublogarithmic-round algorithm for graphs with super-constant girth; i. e. , graphs where the length of the shortest cycle is (1), as shown by Ghaffari~SODA'16. Thus, resolving this 40-year-old open problem requires understanding the family of graphs that contain k-cycles for some constant k. In this work, we come very close to resolving this 40-year-old open problem by presenting a sublogarithmic-round algorithm for graphs that can contain k-cycles for all k > 6. Specifically, our algorithm finds an MIS in O ( (^*) + poly (n) ) rounds, as long as the graph does not contain cycles of length 6, where is the maximum degree of the graph. As a result, we push the limit on the girth of graphs that admit sublogarithmic-round algorithms from k = (1) all the way down to a small constant k=7. This also implies a o (n) round algorithm for MIS in trees, refuting a conjecture from the book by Barrenboim and Elkin.
Khoury et al. (Wed,) studied this question.