Fully Dynamic Maximal Independent Set with Polylogarithmic Update Time

Abstract

We present 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. Our algorithm is randomized and, per update, takes O(log 2 ∆ · log 2 n) expected time. Furthermore, the algorithm can be adjusted to have O(log 2 ∆ · log 4 n) worst-case update-time with high probability. Here, n denotes the number of vertices and ∆ is the maximum degree in the graph. The MIS problem in fully dynamic graphs has attracted significant attention after a breakthrough result of Assadi, Onak, Schieber, and Solomon [STOC'18] who presented an algorithm with O(m 3/4 ) update-time (and thus broke the natural Ω(m) barrier) where m denotes the number of edges in the graph. This result was improved in a series of subsequent papers, though, the update-time remained polynomial. In particular, the fastest algorithm prior to our work had O(min{ √ n, m 1/3 }) update-time [Assadi et al. SODA'19].Our algorithm maintains the lexicographically first MIS over a random order of the vertices. As a result, the same algorithm also maintains a 3-approximation of correlation clustering. We also show that a simpler variant of our algorithm can be used to maintain a random-order lexicographically first maximal matching in the same update-time. * A preliminary version of this paper is

Fully Dynamic Matching: Beating 2-Approximation in Δϵ Update Time

“…Sudan for their collaboration in [6] and Sepehr Assadi for many helpful discussions and pointers to the literature.…”

Random Rank-Based, Hierarchical or Trivial: Which Dynamic Graph Algorithm Performs Best in Practice?

“…We now briefly outline the implemented algorithms for the fully dynamic maximal matching problem. We implement a trivial maximal matching algorithm, as well as the level set partitioning algorithms of Solomon [20] and Baswana et al [2]; and the random rank algorithm of Behnezhad et al [3]. For the algorithm of Baswana et al, we use an implementation of Henzinger et al [12].…”

“…While there exist graph properties, such as singlesource shortest paths distances or maximum flow values in weighted graphs, for which (under some popular complexity assumptions) no sublinear update time is possible [1,13,8], there also exists other graph properties where algorithms with polylogarithmic or even constant update times are known. Such graph properties are connectivity [16,17], minimum spanning tree [17], and maximal independent set [3,7], which all have polylogarithmic time per operation, and maximal matching [20], (∆+1)-vertex coloring [14,5], where ∆ is the maximum degree in the graph, and (1 + )-approximate minimum spanning tree value [15], which all have constant update time. All (non-trivial) dynamic algorithms with polylogarithmic or faster update time used some variant of hierarchical decomposition of a graph into loga-rithmic "layers" [16,17,20,5,15] but in 2019 a new technique, based on giving either each vertex or each edge in the graph a random value of [0, 1], called rank, was introduced into the field independently in three papers [3,7,14].…”

“…(a) For maximal matching we compared the random-rank based algorithm of Behnezhad et al [3] which takes O(log 4 n) amortized update time, the hierarchical algorithm of Solomon [20] which takes O(1) amortized update time, the 2-level version of the hierarchical algorithm of Baswana, Gupta and Sen [2] which takes O( √ n) amortized update time, our own variant of [3] with O(nm) worst-case update time, and a trivial dynamic matching algorithm with O(n) worst-case matching time.…”

Simple dynamic algorithms for Maximal Independent Set, Maximum Flow and Maximum Matching

“…Later, Assadi et al [1] improved it to O(min{∆, m 3/4 }) amortized update time, using a complex case-analysis based algorithm. Allowing randomness this bound was improved to first sublinear in n [2], and then to near constant [10,5]. We present a surprisingly simple algorithm to improve the best deterministic bound for the problem.…”