Fully Dynamic Matching: Beating 2-Approximation in Δ<sup><i>ϵ</i></sup> Update Time

Behnezhad, Soheil; Łącki, Jakub; Mirrokni, Vahab

doi:10.1137/1.9781611975994.152

“…For an unweighted graph, the problem is called Maximum Cardinality Matching, which can be computed in O(m √ n) time [25]. In the dynamic setting, computing α-approximate matching (having cardinality ≥ E M /α) has been extensively studied [4,8,6] with only few results for exact matching [27,28].…”

Section: Notementioning

confidence: 99%

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

Gupta¹,

Khan²

2021

Symposium on Simplicity in Algorithms (SOSA)

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Most graphs in real life keep changing with time. These changes can be in the form of insertion or deletion of edges. Such rapidly changing graphs motivate us to study dynamic graph algorithms. We address three important dynamic graph problems as follows.Maximal Independent Set (MIS) is one of the most prominently studied problems in the distributed setting. Recently, a lot of work studied the dynamic MIS in both deterministic and randomized settings. Allowing randomization the MIS can be maintained in O(poly log n) time under various models [FOCS19], but the best deterministic algorithm [STOC18] requires O(m 3/4 ) amortized update time, which is fairly complex. We present a simple algorithm to improve this to O(m 2/3 ). Additionally, we present simple algorithms to match the lower bound of amortized Ω(n) update time [ICALP16], for maintaining incremental Maximum Flow and Maximum Matching (for unweighted graphs). These algorithms are simple extensions of the incremental reachability algorithm and blossoms algorithm, respectively. Thus, our results either improve an existing upper bound or match the existing lower bounds. A common trait of our results is that they are surprisingly simple, and use no complex data structures or black-box algorithms for their implementation. Further, they are analysed using basic amortization arguments.

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“…Bernstein and Stein [5,6] then showed how to maintain a 3 2 + ǫ approximation in O( 4 √ m/poly(ǫ)) = O( √ n/poly(ǫ)) time-worst-case for bipartite graphs, and amortized for general graphs. In a recent work, Behnezhad et al [4] showed that for any ǫ > 0, there exists a randomized algorithm with worst-case O(∆ ǫ ) = O(n ǫ ) update time, and approximation ratio of 2 − 1 1000•2 13/ǫ , also relying on the search for short augmenting paths. For this, their algorithm crucially relies on the oblivious adversary assumption.…”

Section: Related Workmentioning

confidence: 99%

“…This, however, does not result in o(n) update time for all 2 + ǫ approximation, let alone for 2 − Ω(1) approximation. The only sub-linear time sub-2-approximate algorithms known in general graphs are the aforementioned randomized algorithm of Behnezhad et al [4] and the deterministic amortized time algorithm of Bernstein and Stein [6]. Indeed, for 9/4-approximate matching (or better), no deterministic algorithm for general graphs with o(n) worst-case update time was previously known.…”

Section: Related Workmentioning

confidence: 99%

“…The maximum matching size in B H and B SH are the same as their counterparts without misclassifications, only replacig n f and n i for i ∈ [4] by their tagged counterparts. That is, we have µ(B SH )…”

Section: Approximate Degrees In the Kernelmentioning

confidence: 99%

“…A concentrated effort, starting with the influential work of Onak and Rubinfeld [28], has resulted in numerous algorithms breaking either one of these barriers individually [1-3, 5-7, 9-13, 18, 27-29, 32, 35]. However, despite this long line of work, both barriers were not known to be surpassable simultaneously without assuming sparsity or bipartiteness [5,18,27,29,35], settling for amortized update time [6], or using randomness and the oblivious adversary assumption [4]. Whether or not there exists a deterministic algorithm that beats the trivial folklore O(n)-time maximal matching algorithm both in terms of worst-case update time and approximation ratio, thus simultaneously breaking these two natural barriers for this problem in its full generality, remained a vexing open problem.…”

Section: Introductionmentioning

confidence: 99%

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Beating the Folklore Algorithm for Dynamic Matching

Roghani¹,

Saberi²,

Wajc³

2021

Preprint

0

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The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence 2-approximate) matching in O(n) worst-case update time in n-node graphs.We present the first deterministic algorithm which outperforms the folklore algorithm in terms of both approximation ratio and worst-case update time. Specifically, we give a (2−Ω( 1))-approximate algorithm with O( √ n 8 √ m) = O(n 3/4 ) worst-case update time in n-node, m-edge graphs. For sufficiently small constant ǫ > 0, no deterministic (2 + ǫ)-approximate algorithm with worst-case update time O(n 0.99 ) was known. Our second result is the first deterministic (2 + ǫ)-approximate (weighted) matching algorithm with O ǫ (1) • O( 4 √ m) = O ǫ (1) • O( √ n) worst-case update time.

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Deterministic Dynamic Matching in Worst-Case Update Time

Kiss

¹

2023

Algorithmica

2

0

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We present deterministic algorithms for maintaining a $$(3/2 + \epsilon )$$ ( 3 / 2 + ϵ ) and $$(2 + \epsilon )$$ ( 2 + ϵ ) -approximate maximum matching in a fully dynamic graph with worst-case update times $${\hat{O}}(\sqrt{n})$$ O ^ ( n ) and $${\tilde{O}}(1)$$ O ~ ( 1 ) respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio $$(2 - \delta )$$ ( 2 - δ ) (for any $$\delta > 0$$ δ > 0 ) and $$(2 + \epsilon )$$ ( 2 + ϵ ) were both shown by Roghani et al. (Beating the folklore algorithm for dynamic matching, 2021) with update times $$O(n^{3/4})$$ O ( n 3 / 4 ) and $$O_\epsilon (\sqrt{n})$$ O ϵ ( n ) respectively. We close the gap between worst-case and amortized algorithms for the two approximation ratios as the best deterministic amortized update times for the problem are $$O_\epsilon (\sqrt{n})$$ O ϵ ( n ) and $${\tilde{O}}(1)$$ O ~ ( 1 ) which were shown in Bernstein and Stein (in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, 2016) and Bhattacharya and Kiss (in: 48th international colloquium on automata, languages, and programming, ICALP 2021, 12–16 July, Glasgow, 2021) respectively. The algorithm achieving $$(3/2 + \epsilon )$$ ( 3 / 2 + ϵ ) approximation builds on the EDCS concept introduced by the influential paper of Bernstein and Stein (in: International colloquium on automata, languages, and programming, Springer, Berlin, 2015). Say that H is a $$(\alpha , \delta )$$ ( α , δ ) -approximate matching sparsifier if at all times H satisfies that $$\mu (H) \cdot \alpha + \delta \cdot n \ge \mu (G)$$ μ ( H ) · α + δ · n ≥ μ ( G ) (define $$(\alpha , \delta )$$ ( α , δ ) -approximation similarly for matchings). We show how to maintain a locally damaged version of the EDCS which is a $$(3/2 + \epsilon , \delta )$$ ( 3 / 2 + ϵ , δ ) -approximate matching sparsifier. We further show how to reduce the maintenance of an $$\alpha $$ α -approximate maximum matching to the maintenance of an $$(\alpha , \delta )$$ ( α , δ ) -approximate maximum matching building based on an observation of Assadi et al. (in: Proceedings of the twenty-seventh annual (ACM-SIAM) symposium on discrete algorithms, (SODA) 2016, Arlington, VA, USA, January 10–12, 2016). Our reduction requires an update time blow-up of $${\hat{O}}(1)$$ O ^ ( 1 ) or $${\tilde{O}}(1)$$ O ~ ( 1 ) and is deterministic or randomized against an adaptive adversary respectively. To achieve $$(2 + \epsilon )$$ ( 2 + ϵ ) -approximation we improve on the update time guarantee of an algorithm of Bhattacharya and Kiss (in: 48th International colloquium on automata, languages, and programming, ICALP 2021, 12–16 July, Glasgow, 2021). In order to achieve both results we explicitly state a method implicitly used in Nanongkai and Saranurak (in: Proceedings of the twenty-seventh annual ACM symposium on theory of computing, 2017) and Bernstein et al. (Fully-dynamic graph sparsifiers against an adaptive adversary, 2020) which allows to transform dynamic algorithms capable of processing the input in batches to a dynamic algorithms with worst-case update time.

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

Cited by 14 publications

References 16 publications

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

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

Beating the Folklore Algorithm for Dynamic Matching

Deterministic Dynamic Matching in Worst-Case Update Time

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

Cited by 14 publications

References 16 publications

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

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

Beating the Folklore Algorithm for Dynamic Matching

Deterministic Dynamic Matching in Worst-Case Update Time

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Product

Resources

About

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