Breaking Barriers for Distributed MIS by Faster Degree Reduction

TL;DR

This paper advances distributed algorithms by nearly resolving the open problem of finding MIS in graphs with small cycles, providing a sublogarithmic-round algorithm for graphs with cycles longer than 6, and refuting a prior conjecture.

Contribution

The authors develop a sublogarithmic-round MIS algorithm for graphs containing cycles longer than 6, significantly narrowing the gap in understanding for graphs with small cycles.

Findings

Achieves MIS in $O(rac{ ext{log} \Delta}{ ext{log}( ext{log}^* \Delta)} + ext{poly}( ext{log} ext{log} )$ rounds for certain graphs.

Pushes the girth limit for sublogarithmic algorithms from unbounded to 7.

Provides a $o( ext{sqrt}( ext{log} )$ round MIS algorithm in trees, refuting a previous conjecture.

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 $\omega(1)$, 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 $O\left(\frac{\log \Delta}{\log(\log^* \Delta)} + \mathrm{poly}(\log\log n)\right)$ rounds, as long as the graph does not contain cycles of length $\leq 6$, where $\Delta$ 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 $k = \omega(1)$ 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.