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

Authors:
Sepehr Assadi, Krzysztof Onak, Baruch Schieber,
Shay Solomon.

Abstract:
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(m^{3/4}), 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(m^{2/3}) and O(m^{2/3}√log m), respectively. Du and Zhang also gave a randomized algorithm with update
time O (√m). Moreover, they provided some partial (conditional) hardness results hinting that the update time of m^{1/2−ε}, and in particular n^{1−ε} for n-vertex dense graphs, is a natural barrier
for this problem for any constant ε > 0, for deterministic and randomized algorithms that satisfy a certain natural property.

In this paper, we break this natural barrier and present the first fully dynamic (randomized) algorithm for maintaining an MIS with update time that is always sublinear in the number of vertices, namely, an O (√n) expected amortized update. We also show that a simpler variant of our algorithm can already achieve an O (m^{1/3}) expected amortized update time, which results in an improved performance over our O (√n) update time algorithm for sufficiently sparse graphs, and breaks the m^{1/2} barrier of Du and Zhang for all values of m.

In this paper, we break this natural barrier and present the first fully dynamic (randomized) algorithm for maintaining an MIS with update time that is always sublinear in the number of vertices, namely, an O (√n) expected amortized update. We also show that a simpler variant of our algorithm can already achieve an O (m^{1/3}) expected amortized update time, which results in an improved performance over our O (√n) update time algorithm for sufficiently sparse graphs, and breaks the m^{1/2} barrier of Du and Zhang for all values of m.

Conference version:
[PDF]

Full version:
[arXiv]

BibTex:
[DBLP]