A Distributed (2 + ε)-Approximation for Vertex Cover in O(log Δ / ε log log Δ) Rounds

Bar-Yehuda, Reuven; Censor-Hillel, Keren; Schwartzman, Gregory

doi:10.1145/3060294

Cited by 23 publications

(41 citation statements)

References 27 publications

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“…5 In [23] the same authors argue that these problems have a lower bound of Ω(min{log ∆, √ log n}). However, recently Bar-Yehuda, Censor-Hillel, and Schwartzman [1] pointed our an error in their proof. ∆ 0 .…”

Section: New Resultsmentioning

confidence: 93%

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Chang

Kopelowitz

Pettie

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

184

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Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message:Fast ∆-coloring of trees requires random bits. Building on the recent randomized lower bounds of Brandt et al.[11], we prove that the randomized complexity of ∆-coloring a tree with maximum degree ∆ is Θ(log ∆ log n), for any ∆ ≥ 55, whereas its deterministic complexity is Θ(log ∆ n) for any ∆ ≥ 3. 1 This also establishes a large separation between the deterministic complexity of ∆-coloring and (∆ + 1)-coloring trees.Randomized lower bounds imply deterministic lower bounds. We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(log ∆ n) rounds can be transformed to run in O(log * n − log * ∆ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(log ∆ n) time deterministically. (This gives an alternate proof that deterministically ∆-coloring a tree with small ∆ takes Ω(log ∆ n) rounds.)Deterministic lower bounds imply randomized lower bounds. We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √ log n. This shows that a deterministic Ω(log ∆ n) lower bound for any problem (∆-coloring a tree, for example) implies a randomized Ω(log ∆ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2 O( √ log log n) terms in the complexities of the best MIS and (∆ + 1)-coloring algorithms without also improving the 2 O( √ log n) -round Panconesi-Srinivasan algorithms.

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Section: New Resultsmentioning

confidence: 93%

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Chang

Kopelowitz

Pettie

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

184

View full text Add to dashboard Cite

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“…This is tricky in the distributed environments as we have to coordinate between nodes. For example, the result of [BCS17] does not generalize to hypergraphs, as hyperedges require the coordination of more than two nodes in order to increment edge variables. Our algorithm.…”

Section: Tools and Techniquesmentioning

confidence: 99%

Optimal Distributed Covering Algorithms

Basat

Even

Kawarabayashi

et al. 2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f . This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f . The approximation factor of our algorithm is (f + ε). Let ∆ denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log ∆/ log log ∆) rounds, for constants ε ∈ (0, 1] and f ∈ N + . This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms.For constant values of f and ε, our algorithm improves over the (f + ε)-approximation algorithm of [KMW06] whose running time is O(log ∆ + log W ), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f -approximation for the problem in O(f log n) rounds, improving over the classical result of [KVY94] that achieves a running time of O(f log 2 n). Finally, for weighted vertex cover (f = 2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of [KY11].We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (f + ε)-approximaterounds, where f bounds the number of variables in a constraint, ∆ bounds the number of constraints a variable appears in, and M = max {1, ⌈1/a min ⌉}, where a min is the smallest normalized constraint coefficient. This improves over the results of [KMW06] for the integral case, which combined with rounding achieves the same guarantees in O ε −4 · f 4 · log f · log(M · ∆) rounds.In the Minimum Weight Hypergraph Vertex Cover (mwhvc) problem, we are given a hypergraph G = (V, E) with vertex weights w : V → {1, . . . , W }. 1 The goal is to find a minimum weight cover U ⊆ V such that ∀e ∈ E : e ∩ U = ∅. In this paper we develop a distributed approximation algorithm for mwhvc in the congest model. The approximation ratio is f + ε, where f denotes the rank of the hypergraph (i.e., f is an upper on the size of every hyperedge). The mwhvc problem is a generalization of the Minimum Weight Vertex Cover (mwvc) problem (in which f = 2). The mwhvc problem is also equivalent to the Minimum Weight Set Cover Problem (the rank f of the hypergraph corresponds to the maximum frequency of an element). Both of these problems are among the classical NP-hard problems presented in [Kar72].We consider the following distributed setting for the mwhvc problem. The communication network is a bipartite graph H(E ∪ V, {{e, v} | v ∈ e}). We refer to the network vertices as nodes and network edges as links. The nodes of the network are the hypergraph vertices on one side and hyperedges on the other side. There is a network link between vertex v ∈ V and hyperedge e ∈ E iff v ∈ e. The computation is per...

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“…For clarity of presentation, we first describe an implementation for the LOCAL model. This algorithm can be easily adapted to the CONGEST model using the techniques of [BCS17].…”

Section: A Fast Distributed Implementationmentioning

confidence: 99%

“…(Listing taken from [BCS17] and edited to include our modifications.) 1 γ = parameter in the interval (0, 1).…”

Section: Claim 1 the Following Invariant Holds In Every Iteration Ofmentioning

confidence: 99%

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A Deterministic Distributed 2-Approximation for Weighted Vertex Cover in $$O(\log N\log \varDelta /\log ^2\log \varDelta )$$ Rounds

Ben-Basat

Even

Kawarabayashi³

et al. 2018

Structural Information and Communication Complexity

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We present a deterministic distributed 2-approximation algorithm for the Minimum Weight Vertex Cover problem in the CONGEST model whose round complexity is O(log n log ∆/ log 2 log ∆). This improves over the currently best known deterministic 2-approximation implied by [KVY94]. Our solution generalizes the (2 + ǫ)-approximation algorithm of [BCS17], improving the dependency on ǫ −1 from linear to logarithmic. In addition, for every ǫ = (log ∆) −c , where c ≥ 1 is a constant, our algorithm computes a (2 + ǫ)-approximation in O(log ∆/ log log ∆) rounds (which is asymptotically optimal).

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A Distributed (2 + ε)-Approximation for Vertex Cover in O(log Δ / ε log log Δ) Rounds

Cited by 23 publications

References 27 publications

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Optimal Distributed Covering Algorithms

A Deterministic Distributed 2-Approximation for Weighted Vertex Cover in $$O(\log N\log \varDelta /\log ^2\log \varDelta )$$ Rounds

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