Fully Dynamic MIS in Uniformly Sparse Graphs

Onak, Krzysztof; Schieber, Baruch; Solomon, Shay; Wein, Nicole

doi:10.1145/3378025

Cited by 13 publications

(17 citation statements)

References 29 publications

(34 reference statements)

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“…In such a dynamic setting, recomputing the solution from scratch after every update can be prohibitively time consuming, and it is natural to seek dynamic algorithms that provide faster updates. In the last few decades, efficient dynamic algorithms have been discovered for many combinatorial optimization problems, particularly in graphs such as shortest paths [Fre85,DI04,ACK17a,IKLS17], connectivity [HK99, HDLT01, WN17, ACK17b], maximal independent set and coloring [BCHN18,AOSS18,OSSW18]. For many of these problems, maintaining exact solutions is prohibitively expensive under various complexity conjectures [AW14, KPP16,AWY18,HKNS15a], and thus the best approximation bounds are sought.…”

Section: Introductionmentioning

confidence: 99%

Dynamic set cover: improved algorithms and lower bounds

Abboud¹,

Addanki

Grandoni

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We give new upper and lower bounds for the dynamic set cover problem. First, we give a (1 + ε)f -approximation for fully dynamic set cover in O(f 2 log n/ε 5 ) (amortized) update time, for any ǫ > 0, where f is the maximum number of sets that an element belongs to. In the decremental setting, the update time can be improved to O(f 2 /ε 5 ), while still obtaining an (1 + ε)f -approximation. These are the first algorithms that obtain an approximation factor linear in f for dynamic set cover, thereby almost matching the best bounds known in the offline setting and improving upon the previous best approximation of O(f 2 ) in the dynamic setting.To complement our upper bounds, we also show that a linear dependence of the update time on f is necessary unless we can tolerate much worse approximation factors. Using the recent distributed PCP-framework, we show that any dynamic set cover algorithm that has an amortized update time of O(f 1−ε ) must have an approximation factor that is Ω(n δ ) for some constant δ > 0 under the Strong Exponential Time Hypothesis.

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Section: Introductionmentioning

confidence: 99%

Dynamic set cover: improved algorithms and lower bounds

Abboud¹,

Addanki

Grandoni

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…if the update is insertion then 7 remove v from V k , k > a; 8 run the greedy MIS algorithm on G[S \ {v}] with respect to order π; 9 else 10 add v to all V k , a < k ≤ b; 11 run the greedy MIS algorithm on G[S] with respect to order π; 12 FixSubgraphs(S, b);…”

Section: Correctnessmentioning

confidence: 99%

“…This randomized upper bound was recently improved toÕ( √ n) by Assadi et al [3]. For graphs with bounded arboricity α, a deterministic algorithm with amortized update time of O(α 2 log 2 n) was proposed in [12].…”

Section: Introductionmentioning

confidence: 99%

Fully Dynamic Maximal Independent Set in Expected Poly-Log Update Time

Chechik

Zhang

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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In the fully dynamic maximal independent set (MIS) problem our goal is to maintain an MIS in a given graph G while edges are inserted and deleted from the graph. The first non-trivial algorithm for this problem was presented by Assadi, Onak, Schieber, and Solomon [STOC 2018] who obtained a deterministic fully dynamic MIS with O(m 3/4 ) update time. Later, this was independently improved by Du and Zhang and by Gupta and Khan [arXiv 2018] toÕ(m 2/3 ) update time 1 . Du and Zhang [arXiv 2018] also presented a randomized algorithm against an oblivious adversary withÕ( √ m) update time. The current state of art is by Assadi, Onak, Schieber, and Solomon [SODA 2019] who obtained randomized algorithms against oblivious adversary withÕ( √ n) andÕ(m 1/3 ) update times.In this paper, we propose a dynamic randomized algorithm against oblivious adversary with expected worst-case update time of O(log 4 n). As a direct corollary, one can apply the blackbox reduction from a recent work by Bernstein, Forster, and Henzinger [SODA 2019] to achieve O(log 6 n) worst-case update time with high probability. This is the first dynamic MIS algorithm with very fast update time of poly-log. * Tel Aviv University, shiri.chechik@gmail.com † Tsinghua University, tianyi-z16@mails.tsinghua.edu.cn 1 As usual n is the number of vertices, m is the number of edges andÕ(·) suppresses poly-logarithmic factors.

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“…Later, in a breakthrough, Assadi, Onak, Schieber, and Solomon [5] presented a deterministic algorithm with O(m 3/4 ) update-time; thereby improving the O(m) bound for all graphs. This result was further improved in a series of subsequent papers [23,19,30,6]. The current state-of-the-art is a randomized algorithm due to Assadi et al [6], which requires O(min{ √ n, m 1/3 }) amortized update-time in n-vertex graphs.…”

Section: Introductionmentioning

confidence: 97%

Fully Dynamic Maximal Independent Set with Polylogarithmic Update Time

Behnezhad

Derakhshan

Hajiaghayi

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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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

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Fully Dynamic MIS in Uniformly Sparse Graphs

Cited by 13 publications

References 29 publications

Dynamic set cover: improved algorithms and lower bounds

Dynamic set cover: improved algorithms and lower bounds

Fully Dynamic Maximal Independent Set in Expected Poly-Log Update Time

Fully Dynamic Maximal Independent Set with Polylogarithmic Update Time

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