When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear-Time
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by
Sepehr Assadi, Shay Solomon
2020
Abstract
Maximal independent set (MIS), maximal matching (MM), and
(Δ+1)-coloring in graphs of maximum degree Δ 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 (Δ+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(nlogn·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.
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