# Fully Dynamic Maximal Independent Set with Sublinear in n Update Time

@article{Assadi2018FullyDM,
title={Fully Dynamic Maximal Independent Set with Sublinear in n Update Time},
author={Sepehr Assadi and Krzysztof Onak and Baruch Schieber and Shay Solomon},
journal={ArXiv},
year={2018},
volume={abs/1806.10051}
}
• Published 26 June 2018
• Computer Science, Mathematics
• ArXiv
The first fully dynamic algorithm for maintaining a maximal independent set (MIS) with update time that is sublinear in the number of edges was presented recently by the authors of this paper [Assadi et al., STOC'18]. The algorithm is deterministic and its update time is O(m3/4), where m is the (dynamically changing) number of edges. Subsequently, Gupta and Khan and independently Du and Zhang [arXiv, April 2018] presented deterministic algorithms for dynamic MIS with update times of O(m2/3) and…
• Computer Science
2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
• 2019
The first algorithm for maintaining a maximal independent set (MIS) of a fully dynamic graph---which undergoes both edge insertions and deletions---in polylogarithmic time is presented and a simpler variant of the algorithm can be used to maintain a random-order lexicographically first maximal matching in the same update-time.
• Mathematics, Computer Science
2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
• 2019
A dynamic randomized algorithm against oblivious adversary with expected worst-case update time of O(log^4n).
• Mathematics, Computer Science
ICALP
• 2019
The 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 beta(G) << n.
• Computer Science, Mathematics
ICALP
• 2018
This article significantly improves the update time for uniformly sparse graphs, and improves the result of Assadi et al. (at STOC’18) on maintaining a maximal independent set in a dynamic graph subject to edge insertions and deletions.
• Computer Science, Mathematics
FSTTCS
• 2020
This paper initiates the study of the MAX-CUT problem in fully dynamic graphs. Given a graph G = (V, E), we present deterministic fully dynamic distributed and sequential algorithms to maintain a cut
• Computer Science, Mathematics
ArXiv
• 2020
A framework based on swap operations for resolving the problem to maintain a high-quality independent set, which is an approximation of the MaxIS, over dynamic graphs is presented and two concrete update algorithms based on one- Swappable vertices and two-swappble vertex pairs are designed.
• Computer Science, Mathematics
2022 IEEE 38th International Conference on Data Engineering (ICDE)
• 2022
This paper proposes a framework that maintains a $(\displaystyle \frac{\triangle}{2}+1)$ -approximate MaxIS over dynamic graphs and proves that it achieves a constant approximation ratio in many real-world networks, the first non-trivial approximability result for the dynamic MaxIS problem.
• Computer Science
OPODIS
• 2020
An algorithmic framework for obtaining fast distributed algorithms for a highly-dynamic setting, in which *arbitrarily many* edge changes may occur in each round, and the tasks for which an algorithm is deduced are maximal matching, $(degree+1)$-coloring, 2-approximation for minimum weight vertex cover, and maximal independent set.
• Computer Science
ArXiv
• 2019
A simple algorithm is presented, which handles multiple edge insertions/deletions while keeping the amortized round complexity at a small constant, and it is shown that the same approach can also fix labeling with large labels, given that they can be made to behave as small labels.
• Computer Science
ArXiv
• 2020
This work presents the first deterministic algorithm for maintaining a MIS (and thus a 4-approximate MAX-IS) of a dynamic set of uniform rectangles with amortized sub-logarithmic update time and establishes the trade-off between approximation quality and update time for synthetic and real-world map labeling data sets.

## References

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STOC
• 2018
A deterministic algorithm with amortized update time O(min{Δ,m3/4}), where Δ is a fixed bound on the maximum degree in the graph and m is the (dynamically changing) number of edges.
• Computer Science, Mathematics
ICALP
• 2018
This article significantly improves the update time for uniformly sparse graphs, and improves the result of Assadi et al. (at STOC’18) on maintaining a maximal independent set in a dynamic graph subject to edge insertions and deletions.
• Computer Science, Mathematics
ArXiv
• 2018
A new simple deterministic algorithm with amortized update time of O(m^{2/3}\sqrt{\log m})$is presented, which improves the previous best result and also presents the first randomized algorithm with expected$O(\sqrt{m}\log^{1.5}m)$amortization time against an oblivious adversary. • Computer Science ArXiv • 2018 A surprisingly simple deterministic centralized algorithm which improves the amortized update time to$O(\min\{\Delta,m^{2/3}\})\$ and some other minor results related to dynamic MIS, Maximum Flow, and Maximum Matching are presented.
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STOC
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It is shown that a conjecture that there is no truly subcubic (O(n3-ε) time algorithm for this problem can be used to exhibit the underlying polynomial time hardness shared by many dynamic problems.
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PODC
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This paper shows how to update an MIS in a dynamic distributed setting, either synchronous or asynchronous, with only a single adjustment and in a single round, in expectation, which strongly separates the static and dynamic distributed models.
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SODA
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The technique can be used to simplify and significantly speed up the preprocessing time for the emergency planning problem while matching previous bounds for an update, and to approximate the sizes of cutsets of dynamic graphs in time O(min{|S|, |V\S|}) for an oblivious adversary.
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• Computer Science, Mathematics
28th Annual Symposium on Foundations of Computer Science (sfcs 1987)
• 1987
A maximal independent set in an n-cycle cannot be found faster than Ω(log* n) and this is optimal by [CV]; the d-regular tree of radius r cannot be colored with fewer than √d colors in time 2r / 3.