## When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear Time

Authors: Sepehr Assadi, Shay Solomon
Abstract: Maximal independent set (MIS), maximal matching (MM), and (∆+ 1)-coloring in graphs of maximum degree ∆ are among the most prominent algorithmic graph theory problems. They are all solvable by a simple linear-time greedy algorithm and up until very recently this constituted the state-of-the-art. In SODA 2019, Assadi, Chen, and Khanna gave a randomized algorithm for (∆ + 1)-coloring that runs in ~O(n√n) time, which even for moderately dense graphs is sublinear in the input size. The work of Assadi et al. however contained a spoiler for MIS and MM: neither problems provably admits a sublinear-time algorithm in general graphs. In this work, we dig deeper into the possibility of achieving sublinear-time algorithms for MIS and MM.

The neighborhood independence number of a graph G, denoted by β(G), is the size of the largest independent set in the neighborhood of any vertex. We identify β(G) as the “right” parameter to measure the runtime of MIS and MM algorithms: Although graphs of bounded neighborhood independence may be very dense (clique is one example), we prove that carefully chosen variants of greedy algorithms for MIS and MM run in O(nβ(G)) and O(n log n · β(G)) time respectively on any n-vertex graph G. We complement this positive result by observing that a simple extension of the lower bound of Assadi et al. implies that Ω(nβ(G)) time is also necessary for any algorithm to either problem for all values of β(G) from 1 to Θ(n). We note that our algorithm for MIS is deterministic while for MM we use randomization which we prove is unavoidable: any deterministic algorithm for MM requires Ω(n^2) time even for β(G) = 2.

Graphs with bounded neighborhood independence, already for constant β = β(G), constitute a rich family of possibly dense graphs, including line graphs, proper interval graphs, unit-disk graphs, claw-free graphs, and graphs of bounded growth. Our results suggest that even though MIS and MM do not admit sublinear-time algorithms in general graphs, one can still solve both problems in sublinear time for a wide range of β(G) ≪ n. Finally, by observing that the lower bound of Ω(n√n) time for (∆+1)-coloring due to Assadi et al. applies to graphs of (small) constant neighborhood independence, we unveil an intriguing separation between the time complexity of MIS and MM, and that of (∆ + 1)-coloring: while the time complexity of MIS and MM is strictly higher than that of (∆ + 1) coloring in general graphs, the exact opposite relation holds for graphs with small neighborhood independence.
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