Breaking Barriers for Distributed MIS by Faster Degree Reduction
Seri Khoury, Aaron Schild
TL;DR
The paper investigates the randomized MIS problem in the LOCAL model, where the classical $O(\log n)$ bounds have stood for decades. It introduces a novel two-round MIS strategy that achieves a higher per-round inclusion probability, enabling faster degree reduction and a post-shattering framework that partitions the graph into tractable parts. For graphs with girth at least $7$ (no cycles of length $\le 6$), it proves MIS in $O\left(\frac{\log \Delta}{\log(\log^* \Delta)} + \mathrm{poly}(\log\log n)\right)$ rounds, and derives consequences such as $o(\sqrt{\log n})$-round MIS in trees, along with a separation between MIS and Maximal Matching in trees. Collectively, these results push the boundary toward resolving the central open question of whether MIS can be achieved in sublogarithmic time on general graphs, and they illuminate new barriers and techniques (higher inclusion probability, two-round progression, and shattering) applicable to broader distributed graph problems.
Abstract
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 $O(\log n)$ rounds in $n$-node graphs with high probability. Despite decades of research, the existence of any $o(\log n)$-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 $ω(1)$, as shown by Ghaffari~[SODA'16]. Thus, resolving this $\approx 40$-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 $\approx 40$-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 $O\left(\frac{\log Δ}{\log(\log^* Δ)} + \mathrm{poly}(\log\log n)\right)$ rounds, as long as the graph does not contain cycles of length $\leq 6$, where $Δ$ 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 $k = ω(1)$ all the way down to a small constant $k=7$. This also implies a $o(\sqrt{\log n})$ round algorithm for MIS in trees, refuting a conjecture from the book by Barrenboim and Elkin.