Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching

Bhattacharya, Sayan; Henzinger, Monika; Italiano, Giuseppe F.

doi:10.1137/140998925

Cited by 36 publications

(52 citation statements)

References 12 publications

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“…We believe that this is an important question, since it may help in understanding how to develop deterministic dynamic algorithms in general. It is very challenging and interesting to design deterministic dynamic algorithms with performances similar to the randomized ones for many dynamic graph problems such as maximal matching [5,10,9,11,12,13], connectivity [24,28,32,27], and shortest paths [6,8,7,21,22].…”

Section: Open Problemsmentioning

confidence: 99%

Dynamic Algorithms for Graph Coloring

Bhattacharya¹,

Chakrabarty²,

Henzinger³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for (∆ + 1)-vertex coloring and (2∆ − 1)-edge coloring in a graph with maximum degree ∆. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results.(1) We present a randomized algorithm which maintains a (∆ + 1)-vertex coloring with O(log ∆) expected amortized update time. (2) We present a deterministic algorithm which maintains a (1 + o (1)

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Section: Open Problemsmentioning

confidence: 99%

Dynamic Algorithms for Graph Coloring

Bhattacharya¹,

Chakrabarty²,

Henzinger³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…if v = i then // At this state, v should be matched in M 0 through its incident edge e. 12 if is_sampled i (e) = 0 then 13 Vertex v is not in the vertex-set of graph G i . If v is the lower-ID endpoint of e,…”

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confidence: 99%

Fully Dynamic Matching: Beating 2-Approximation in Δ^ϵ Update Time

Behnezhad¹,

Łącki²,

Mirrokni³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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In fully dynamic graphs, we know how to maintain a 2-approximation of maximum matching extremely fast, that is, in polylogarithmic update time or better. In a sharp contrast and despite extensive studies, all known algorithms that maintain a 2−Ω(1) approximate matching are much slower. Understanding this gap and, in particular, determining the best possible update time for algorithms providing a better-than-2 approximate matching is a major open question.In this paper, we show that for any constant ε > 0, there is a randomized algorithm that with high probability maintains a 2 − Ω(1) approximate maximum matching of a fully-dynamic general graph in worst-case update time O(∆ ε + polylog n), where ∆ is the maximum degree.Previously, the fastest fully dynamic matching algorithm providing a better-than-2 approximation had O(m 1/4 ) update-time [Bernstein and Stein, SODA 2016]. A faster algorithm with update-time O(n ε ) was known, but worked only for maintaining the size (and not the edges) of the matching in bipartite graphs [Bhattacharya, Henzinger, and Nanongkai, STOC 2016].

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“…1 Note that dynamic set cover as defined above is a generalization of the dynamic vertex cover problem which, together with the dynamic maximum matching problem, have been studied extensively in recent years (e.g. [31,3,20,13,9,10,12,35,30,32,18,6,7,8,33,36,2,25]).…”

Section: Delete(e)mentioning

confidence: 99%

A New Deterministic Algorithm for Dynamic Set Cover

Bhattacharya

Henzinger

Nanongkai

2019

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

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We present a deterministic dynamic algorithm for maintaining a (1+ǫ)f -approximate minimum cost set cover with O(f log(Cn)/ǫ 2 ) amortized update time, when the input set system is undergoing element insertions and deletions. Here, n denotes the number of elements, each element appears in at most f sets, and the cost of each set lies in the range [1/C, 1]. Our result, together with that of Gupta et al. [STOC'17], implies that there is a deterministic algorithm for this problem with O(f log(Cn)) amortized update time and O(min(log n, f ))-approximation ratio, which nearly matches the polynomial-time hardness of approximation for minimum set cover in the static setting. Our update time is only O(log(Cn)) away from a trivial lower bound.Prior to our work, the previous best approximation ratio guaranteed by deterministic algorithms was O(f 2 ), which was due to Bhattacharya et al. [ICALP'15]. In contrast, the only result that guaranteed O(f )-approximation was obtained very recently by Abboud et al. [STOC'19], who designed a dynamic algorithm with (1+ǫ)f -approximation ratio and O(f 2 log n/ǫ) amortized update time. Besides the extra O(f ) factor in the update time compared to our and Gupta et al.'s results, the Abboud et al. algorithm is randomized, and works only when the adversary is oblivious and the sets are unweighted (each set has the same cost).We achieve our result via the primal-dual approach, by maintaining a fractional packing solution as a dual certificate. This approach was pursued previously by Bhattacharya et al. and Gupta et al., but not in the recent paper by Abboud et al. Unlike previous primal-dual algorithms that try to satisfy some local constraints for individual sets at all time, our algorithm basically waits until the dual solution changes significantly globally, and fixes the solution only where the fix is needed.

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Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching

Cited by 36 publications

References 12 publications

Dynamic Algorithms for Graph Coloring

Dynamic Algorithms for Graph Coloring

Fully Dynamic Matching: Beating 2-Approximation in Δ^ϵ Update Time

A New Deterministic Algorithm for Dynamic Set Cover

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Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching

Cited by 36 publications

References 12 publications

Dynamic Algorithms for Graph Coloring

Dynamic Algorithms for Graph Coloring

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

A New Deterministic Algorithm for Dynamic Set Cover

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Fully Dynamic Matching: Beating 2-Approximation in Δ^ϵ Update Time