Deterministic Distributed Dominating Set Approximation in the CONGEST Model

Deurer, Janosch; Kühn, Fabian; Maus, Yannic

doi:10.1145/3293611.3331626

Cited by 26 publications

(18 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More recently, Deurer, Kuhn, and Maus [DKM19] gave a stronger result for minimum dominating set. They obtain a 2 O( √ log n•log log n) round deterministic distributed algorithm that computes a O(log ∆) approximation of minimum dominating set.…”

Section: B Sparse Neighborhood Covermentioning

confidence: 99%

“…Other Problems: Spanners and Dominating Set Approximation Due to previous applications of k-hop separated network decompositions by Ghaffari and Kuhn [GK18] as well as Deurer, Kuhn, and, Maus [DKM19] we obtain the following deterministic CONGEST model algorithms: In Appendix C, we review a 2 O( √ log n) round algorithm for computing a (2k − 1)-stretch spanner with size O(kn 1+1/k log n), and a O(log ∆)-approximation algorithm for minimum dominating set in 2 O( √ log n) rounds.…”

Section: Applications: Mis Neighborhood Cover and Beyondmentioning

confidence: 99%

See 1 more Smart Citation

Improved Network Decompositions using Small Messages with Applications on MIS, Neighborhood Covers, and Beyond

Ghaffari¹,

Portmann²

2019

Preprint

View full text Add to dashboard Cite

Network decompositions, as introduced by Awerbuch, Luby, Goldberg, and Plotkin [FOCS'89], are one of the key algorithmic tools in distributed graph algorithms. We present an improved deterministic distributed algorithm for constructing network decompositions of power graphs using small messages, which improves upon the algorithm of Ghaffari and Kuhn [DISC'18]. In addition, we provide a randomized distributed network decomposition algorithm, based on our deterministic algorithm, with failure probability exponentially small in the input size that works with small messages as well. Compared to the previous algorithm of Elkin and Neiman [PODC'16], our algorithm achieves a better success probability at the expense of its round complexity, while giving a network decomposition of the same quality. As a consequence of the randomized algorithm for network decomposition, we get a faster randomized algorithm for computing a Maximal Independent Set, improving on a result of Ghaffari [SODA'19]. Other implications of our improved deterministic network decomposition algorithm are: a faster deterministic distributed algorithms for constructing spanners and approximations of distributed set cover, improving results of Ghaffari, and Kuhn [DISC'18] and Deurer, Kuhn, and Maus [PODC'19]; and faster a deterministic distributed algorithm for constructing neighborhood covers, resolving an open question of Elkin [SODA'04].CONGEST model, Ghaffari and Kuhn [GK18] showed that k-hop separated network decompositions can be used for computing spanners and approximating minimum dominating set. State of the Art-DeterministicConstructions: There are four known deterministic distributed constructions of network decompositions, successively improving either quantitatively or qualitatively [ALGP89, PS92, GK18, Gha19]. Awerbuch et al. [ALGP89] provided an algorithm for computing (2 O( √ log n log log n) , 2 O( √ log n log log n) ) network decompositions of an n node graph G in 2 O( √ log n log log n) rounds, which works in the CONGEST model. Subsequently, this was improved by Panconesi and Srinivasan [PS92] showing that all 2 O( √ log n log log n) terms could be replaced by 2 O( √ log n). However, their algorithm requires large messages. For network decompositions with higher levels of separation, Ghaffari and Kuhnnetwork decomposition of G k , which works with small messages. Note that extending network decomposition algorithms to compute a decomposition of G k is trivial in the LOCAL model: As nodes can send messages of arbitrary size, communication on G k can be simulate in k rounds of communication on G. Thus, with a k factor overhead in the round complexity (and a k factor increase in the diameter with respect to distances in G), we can use any LOCAL-model network decomposition algorithm to also compute k-hop separated decompositions.Recently, Ghaffarinetwork decomposition can also be computed in 2 O( √ log n) rounds in the CONGEST model. However, his construction cannot extend to G k , which is one of the issues we address in this paper. In con...

show abstract

Section: B Sparse Neighborhood Covermentioning

confidence: 99%

Section: Applications: Mis Neighborhood Cover and Beyondmentioning

confidence: 99%

Improved Network Decompositions using Small Messages with Applications on MIS, Neighborhood Covers, and Beyond

Ghaffari¹,

Portmann²

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…We also note that there are also similar efficient deterministic CONGESTmodel algorithms for other problems, including Δ + 1 coloring (and even degree + 1 list coloring), and dominating set and set cover approximations. See [45,Corollary 3.17 & 3.18] and [16,26]. All of these rely on explicit derandomization of some randomized algorithm.…”

Section: Open Problems: Congest Modelmentioning

confidence: 99%

Network Decomposition and Distributed Derandomization (Invited Paper)

Ghaffari

2020

Structural Information and Communication Complexity

View full text Add to dashboard Cite

We overview a recent line of work [Rozhoň and Ghaffari at STOC 2020; Ghaffari, Harris, and Kuhn at FOCS 2018; and Ghaffari, Kuhn, and Maus at STOC 2017], which proved that any (locallycheckable) graph problem that admits an efficient randomized distributed algorithm also admits an efficient deterministic distributed algorithm, thereby resolving several central and decades-old open problems in distributed graph algorithms. We present a short and self-contained version of the proofs, and conclude by discussing several related questions that remain open.This article accompanies a keynote talk of the author at the International Colloquium on Structural Information and Communication Complexity (SIROCCO) 2020. The writing is based on [24,28,45] and primarily targets non-experts.

show abstract

“…So far, there are a few examples where a small diameter helps in the CONGEST model. In particular, the problems of computing a maximal independent set, a sparse spanner, and an (1 + ǫ) log ∆-approximation of a minimum dominating set can all be solved deterministically in time D • polylog n [CPS17, GK18,DKM19]. Other problems cannot profit from small diameter, e.g., verifying or computing a minimum spanning tree requires Ω(D + √ n) CONGEST rounds [DHK + 11] and solving many optimization problems exactly (minimum dominating set, vertex cover, chromatic number) or almost exactly (maximum independent set) requires Ω(n 2 ) CONGEST rounds even if the diameter of the graph is constant [CKP17, BCD + 19].…”

Section: Our Contributions In Congestmentioning

confidence: 99%

Efficient Deterministic Distributed Coloring with Small Bandwidth

Bamberger¹,

Kühn²,

Maus³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We show that the (degree + 1)-list coloring problem can be solved deterministically in O(D • log n • log 2 ∆) rounds in the CONGEST model, where D is the diameter of the graph, n the number of nodes, and ∆ the maximum degree. Using the recent polylogarithmic-time deterministic network decomposition algorithm by Rozhoň and Ghaffari [STOC 2020], this implies the first efficient (i.e., poly log n-time) deterministic CONGEST algorithm for the (∆+1)coloring and the (degree+1)-list coloring problem. Previously the best known algorithm required 2 O( √ log n) rounds and was not based on network decompositions. Our techniques also lead to deterministic (degree + 1)-list coloring algorithms for the congested clique and the massively parallel computation (MPC) model. For the congested clique, we obtain an algorithm with time complexity O(log ∆•log log ∆), for the MPC model, we obtain algorithms with round complexity O(log 2 ∆) for the linear-memory regime and O(log 2 ∆+log n) for the sublinear memory regime.Distributed Graph Coloring. Computing a coloring with the optimal number of colors χ(G) was one of the first problems known to be NP-complete [Kar72]. In the distributed setting, one therefore aims for a more relaxed goal and the usual objective is to color a given graph G with ∆ + 1 colors, where ∆ is the maximum degree of G [BE13]. Note that any graph admits such a coloring and it can be computed by a simple sequential greedy algorithm. Despite the simplicity of the sequential algorithm, the question of determining the distributed complexity of computing a (∆ + 1)-coloring has been an extremely challenging question. In particular, while O(log n)-time randomized distributed (∆+1)-coloring algorithms have been known for more than 30 years, the question whether a similarly efficient (i.e., polylogarithmic time) deterministic distributed coloring algorithm exists remained one of the most important open problems in the area [BE13, GKM17] until it was resolved very recently by Rozhoň and Ghaffari [RG19]. In [RG19] the question was answered in the affirmative for the LOCAL model by designing an efficient deterministic distributed method to decompose the communication graph into a logarithmic number of subgraphs consisting of connected components (clusters) of polylogarithmic (weak) diameter, a structure known as a network decomposition [AGLP89]. It was known before that an efficient deterministic algorithm for network decomposition would essentially imply efficient determinstic LOCAL algorithms for all problems for which efficient randomized algorithms exist [Lin92, AGLP89, BE13, GKM17, GHK18].Note that due to the unbounded message size any (solvable) problem can be solved in diameter time in the LOCAL model by simply collecting the whole graph topology at one node, solving the problem locally (potentially using unbounded computational power), and redistributing the solution to the nodes. Thus the small diameter components of a network decomposition almost immediately give rise to an efficient (∆ + 1)coloring algorithm in the...

show abstract

Deterministic Distributed Dominating Set Approximation in the CONGEST Model

Cited by 26 publications

References 42 publications

Improved Network Decompositions using Small Messages with Applications on MIS, Neighborhood Covers, and Beyond

Improved Network Decompositions using Small Messages with Applications on MIS, Neighborhood Covers, and Beyond

Network Decomposition and Distributed Derandomization (Invited Paper)

Efficient Deterministic Distributed Coloring with Small Bandwidth

Contact Info

Product

Resources

About