Distributed Local Approximation Algorithms for Maximum Matching in Graphs and Hypergraphs

Harris, David G.

doi:10.1109/focs.2019.00048

Cited by 19 publications

(6 citation statements)

References 33 publications

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“…By a reduction that they provided from (2∆−1)-edge coloring in graphs to maximal match-ing in hypergraphs of rank 3, this led to a poly log n-time deterministic algorithm for (2∆ − 1)-edge coloring, hence putting the second problem in the poly(log n) regime. Harris [Har19] improved the complexity to O(log 2 ∆ • log n).…”

Section: State Of the Artmentioning

confidence: 99%

Faster Deterministic Distributed MIS and Approximate Matching

Ghaffari¹,

Grunau²

2023

Preprint

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We present an O(log 2 n) round deterministic distributed algorithm for the maximal independent set problem. By known reductions, this round complexity extends also to maximal matching, ∆ + 1 vertex coloring, and 2∆ − 1 edge coloring. These four problems are among the most central problems in distributed graph algorithms and have been studied extensively for the past four decades. This improved round complexity comes closer to the Ω(log n) lower bound of maximal independent set and maximal matching [Balliu et al. FOCS '19]. The previous best known deterministic complexity for all of these problems was Θ(log 3 n). Via the shattering technique, the improvement permeates also to the corresponding randomized complexities, e.g., the new randomized complexity of ∆ + 1 vertex coloring is now O(log 2 log n) rounds.Our approach is a novel combination of the previously known (and seemingly orthogonal) two methods for developing fast deterministic algorithms for these problems, namely global derandomization via network decomposition (see e.g., [Rozhon, Ghaffari STOC'20; Ghaffari, Grunau, Rozhon SODA'21; Ghaffari et al. SODA'23]) and local rounding of fractional solutions (see e.g., [Fischer DISC'17; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Ghaffari, Kuhn FOCS'21; Faour et al. SODA'23]). We consider a relaxation of the classic network decomposition concept, where instead of requiring the clusters in the same block to be non-adjacent, we allow each node to have a small number of neighboring clusters. We also show a deterministic algorithm that computes this relaxed decomposition faster than standard decompositions. We then use this relaxed decomposition to significantly improve the integrality of certain fractional solutions, before handing them to the local rounding procedure that now has to do fewer rounding steps.Randomized algorithms, and the pursuit of deterministic algorithms. In the 1980s, Luby [Lub86] and Alon, Babai, and Itai [ABI86] presented a simple and elegant randomized distributed algorithm that computes an MIS in O(log n) rounds, with high probability 1 . Due to known reductions, this MIS algorithm led to O(log n) round randomized algorithms for many other key graph problems, including maximal matching, ∆ + 1 vertex coloring, and (2∆ − 1) edge coloring. These problems are often listed as the four fundamental symmetry-breaking problems in distributed graph algorithms and have a wide range of applications. The O(log n)-round randomized algorithm naturally led the researchers to seek a deterministic distributed algorithm with the same round complexity. In his celebrated work [Lin87, Lin92], Linial asked "can it [MIS] always be found in polylogarithmic time [deterministically]? " He even added that "getting a deterministic polylog-time algorithm for MIS seems hard." Since then, this became known as Linial's MIS question and turned into one of the research foci in distributed graph algorithms.The two approaches of deterministic algorithms. Linial's MIS problem remained open for nearly three decades. During...

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Section: State Of the Artmentioning

confidence: 99%

Faster Deterministic Distributed MIS and Approximate Matching

Ghaffari¹,

Grunau²

2023

Preprint

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“…As mentioned, the ability to color greedily makes 2∆ − 1 edge coloring particularly amenable to distributed algorithms, and a randomized procedure of Elkin, Pettie and Schneider [13] combined with a deterministic algorithm of Fischer, Ghaffari, and Kuhn [16] yielded the first poly-log-logarithmic randomized round complexity of O(log 9 log n) for the problem. This round complexity was later improved to O(log 6 log n) [19] and then to Õ(log 3 log n) [23]. Very recently, a poly(log ∆) + O(log * n)-round deterministic algorithm was also given for the problem [2].…”

Section: Edge Coloringmentioning

confidence: 99%

“…The picture for edge coloring is also far from complete: there remain large gaps between upper and lower bounds on distributed complexity for most palette size regimes. For 2∆ − 1-coloring, log O (1) log nround algorithms are known [16,19,23], but the only lower bound is Ω(log * n) [26,30]. Below 2∆ − 1 colors, there is an Ω(log ∆ log n)-round lower bound [9], and Corollary 3.1 closes the corresponding upper bound to only a polynomial gap for some ∆ + o(∆) number of colors 6 .…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

Improved Distributed Algorithms for the Lovász Local Lemma and Edge Coloring

Davies

2023

Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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The Lovász Local Lemma is a classic result in probability theory that is often used to prove the existence of combinatorial objects via the probabilistic method. In its simplest form, it states that if we have n 'bad events', each of which occurs with probability at most p and is independent of all but d other events, then under certain criteria on p and d, all of the bad events can be avoided with positive probability.While the original proof was existential, there has been much study on the algorithmic Lovász Local Lemma: that is, designing an algorithm which finds an assignment of the underlying random variables such that all the bad events are indeed avoided. Notably, the celebrated result of Moser and Tardos [JACM '10] also implied an efficient distributed algorithm for the problem, running in O(log 2 n) rounds. For instances with low d, this was improved to O(d 2 + log O(1) log n) by Fischer and Ghaffari [DISC '17], a result that has proven highly important in distributed complexity theory (Chang and Pettie [SICOMP '19]).We give an improved algorithm for the Lovász Local Lemma, providing a trade-off between the strength of the criterion relating p and d, and the distributed round complexity. In particular, in the same regime as Fischer and Ghaffari's algorithm, we improve the round complexity to O( d log d + log O(1) log n). At the other end of the trade-off, we obtain a log O(1) log n round complexity for a substantially wider regime than previously known.As our main application, we also give the first log O(1) log n-round distributed algorithm for the problem of ∆ + o(∆)-edge coloring a graph of maximum degree ∆. This is an almost exponential improvement over previous results: no prior log o(1) n-round algorithm was known even for 2∆ − 2-edge coloring.

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“…Previously, the best deterministic algorithm for general graphs had a time complexity of 2 O( √ log n) and as in the case of the best (∆ + 1)-vertex coloring algorithm, it was a brute-force algorithm based on first computing a network decomposition [AGLP89,PS95b]. Together with the best known randomized algorithms (which use the shattering technique), the algorithm of [FGK17] also implied that the (2∆ − 1)-edge coloring problem has a randomized complexity of poly log log n. The current best time complexities for computing an (2∆ − 1)-edge coloring in general graphs areÕ(log 2 ∆ log n) for the deterministic andÕ(log 3 log n) for the randomized setting [Har18]. For small values of ∆, the best known complexity is O( √ ∆ log ∆ log * ∆+log * n) [FHK16,BEG18], as for computing a (∆+1)-vertex coloring.…”

Section: Randomized Distributedmentioning

confidence: 99%

Faster Deterministic Distributed Coloring Through Recursive List Coloring

Kühn¹

2020

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

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We provide novel deterministic distributed vertex coloring algorithms. As our main result, we give a deterministic distributed algorithm to compute a (∆ + 1)-coloring of an n-node graph with maximum degree ∆ in 2 O( √ log ∆) · log n rounds. This improves on the best previously known time complexity of min O √ ∆ log ∆ log * ∆ + log * n , 2 O( √ log n) for a large range of values of ∆. For graphs with arboricity a, we obtain a deterministic distributed algorithm to compute a (2+o(1))acoloring in time 2 O( √ log a) · log 2 n. Further, for graphs with bounded neighborhood independence, we show that a (∆ + 1)-coloring can be computed more efficiently in time 2 O(This in particular implies that also a (2∆ − 1)-edge coloring can be computed deterministically in 2 O( √ log ∆) + O(log * n) rounds, which improves the best known time bound for small values of ∆. All results even hold for the list coloring variants of the problems. As a consequence, we also obtain an improved deterministic 2 O( √ log ∆) · log 3 n-round algorithm for ∆-coloring non-complete graphs with maximum degree ∆ ≥ 3. Most of our algorithms only require messages of O(log n) bits (including the (∆ + 1)-vertex coloring algorithms).Our main technical contribution is a recursive deterministic distributed list coloring algorithm to solve list coloring problems with lists of size ∆ 1+o(1) . Given some list coloring problem and an orientation of the edges, we show how to recursively divide the global color space into smaller subspaces, assign one of the subspaces to each node of the graph, and compute a new edge orientation such that for each node, the list size to out-degree ratio degrades at most by a constant factor on each recursion level.

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Distributed Local Approximation Algorithms for Maximum Matching in Graphs and Hypergraphs

Cited by 19 publications

References 33 publications

Faster Deterministic Distributed MIS and Approximate Matching

Faster Deterministic Distributed MIS and Approximate Matching

Improved Distributed Algorithms for the Lovász Local Lemma and Edge Coloring

Faster Deterministic Distributed Coloring Through Recursive List Coloring

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