Abstract
We consider the problem of maintaining a maximal independent set in a dynamic graph subject to edge insertions and deletions. Recently, Assadi et al. (at STOC’18) showed that a maximal independent set can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this article, we significantly improve the update time for uniformly sparse graphs . Specifically, for graphs with arboricity α, the amortized update time of our algorithm is O (α 2 ⋅ log 2 n ), where n is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs and some classes of “real-world” graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by m 3/8−ϵ , for any constant ϵ > 0. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed m 1/2 .
Highlights
The importance of the maximal independent set (MIS) problem is hard to overstate
MIS is often used in the context of graph coloring, as all vertices in an independent set can be assigned the same color
In the 1980s, questions concerning the computational complexity of the MIS problem spurred a line of research that led to the celebrated parallel algorithms of Luby [18], and Alon, Babai, and Itai [1]
Summary
The importance of the maximal independent set (MIS) problem is hard to overstate. In general, MIS algorithms constitute a useful subroutine for locally breaking symmetry between several choices. In the 1980s, questions concerning the computational complexity of the MIS problem spurred a line of research that led to the celebrated parallel algorithms of Luby [18], and Alon, Babai, and Itai [1]. In STOC’18, Assadi, Onak, Schieber, and Solomon [2] gave the first sub-linear (amortized) update time fully dynamic algorithm for maintaining a MIS. Their amortized update time in min{m3/4, ∆}, where m is the (dynamically changing) number of edges, and ∆ is a fixed bound on the maximum degree of the graph. For graphs of high maximum degree, the update time of the algorithm of [2] decreases as the graph becomes sparser
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