Exponential Energy Savings in Local Distributed Graph Algorithms

Authors: Ghaffari, Mohsen, Yeoh, Zi Song

Source: Proceedings of the 38th ACM Symposium on Parallelism in Algorithms and Architectures. :261-287

Availability: http://dl.acm.org/doi/10.1145/3816782.3819215

Distributed Algorithms for Potential Problems

Authors: Balliu, Alkida, Boudier, Thomas, d'Amore, Francesco, Kuhn, Fabian, Olivetti, Dennis, Schmid, Gustav, Suomela, Jukka

Source: ACM Symposium on Principles of Distributed Computing. :154-165

Availability: http://dl.acm.org/doi/10.1145/3796701.3815928

Distributed Approximate Maximum Matching and Minimum Vertex Cover via Generalized Graph Decomposition

Authors: Davies-Peck, Peter

Source: ACM Symposium on Principles of Distributed Computing. :371-381

Availability: http://dl.acm.org/doi/10.1145/3796701.3815909

Towards Optimal Distributed Delta Coloring

Additional Titles: The $Δ$-vertex coloring problem has become one of the prototypical problems for understanding the complexity of local distributed graph problems on constant-degree graphs. The major open problem is whether the problem can be solved deterministically in logarithmic time, which would match the lower bound [Chang et al., FOCS'16]. Despite recent progress in the design of efficient $Δ$-coloring algorithms, there is currently a polynomial gap between the upper and lower bounds. In this work we present a $O(\log n)$-round deterministic $Δ$-coloring algorithm for dense constant-degree graphs, matching the lower bound for the problem on general graphs. For general $Δ$ the algorithms' complexity is $\min\{\widetilde{O}(\log^{5/3}n),O(Δ+\log n)\}$. All recent distributed and sublinear graph coloring algorithms (also for coloring with more than $Δ$ colors) decompose the graph into sparse and dense parts. Our algorithm works for the case that this decomposition has no sparse vertices. Ironically, in recent (randomized) $Δ$-coloring algorithms, dealing with sparse parts was relatively easy and these dense parts arguably posed the major hurdle. We present a solution that addresses the dense parts and may have the potential for extension to sparse parts. Our approach is fundamentally different from prior deterministic algorithms and hence hopefully contributes towards designing an optimal algorithm for the general case. Additionally, we leverage our result to also obtain a randomized $\min\{\widetilde{O}(\log^{5/3}\log n), O(Δ+\log\log n)\}$-round algorithm for $Δ$-coloring dense graphs that also matches the lower bound for the problem on general constant-degree graphs [Brandt et al.; STOC'16].

Authors: Jakob, Manuel, Maus, Yannic

Index Terms: Distributed, Parallel, and Cluster Computing, Data Structures and Algorithms, text

URL: http://arxiv.org/abs/2504.03080

On the Complexity of Distributed Edge Coloring and Orientation Problems

Additional Titles: Understanding the role of randomness when solving locally checkable labeling (LCL) problems in the LOCAL model has been one of the top priorities in the research on distributed graph algorithms in recent years. For LCL problems in bounded-degree graphs, it is known that randomness cannot help more than polynomially, except in one case: if the deterministic complexity of an LCL problem is in $Ω(\log n)$ and its randomized complexity is in $o(\log n)$, then the randomized complexity is guaranteed to be $poly(\log \log n)$. However, the fundamental question of \emph{which} problems with a deterministic complexity of $Ω(\log n)$ can be solved exponentially faster using randomization still remains wide open. We make a step towards answering this question by studying a simple, but natural class of LCL problems: so-called degree splitting problems. These problems come in two varieties: coloring problems where the edges of a graph have to be colored with $2$ colors and orientation problems where each edge needs to be oriented. For $Δ$-regular graphs (where $Δ=O(1)$), we obtain the following results. - We gave an exact characterization of the randomized complexity of all problems where the edges need to be colored with two colors, say red and blue, and which have a deterministic complexity of $O(\log n)$. - For edge orientation problems, we give a partial characterization of the problems that have a randomized complexity of $Ω(\log n)$ and the problems that have a randomized complexity of $poly\log\log n$. While our results are cleanest to state for the $Δ$-regular case, all our algorithms naturally generalize to nodes of any degree $d<Δ$ in general graphs of maximum degree $Δ$.

Authors: Brandt, Sebastian, Kuhn, Fabian, Parsaeian, Zahra

Index Terms: Data Structures and Algorithms, text

URL: http://arxiv.org/abs/2510.21327

The Distributed Complexity of Locally Checkable Labeling Problems Beyond Paths and Trees

Additional Titles: We consider locally checkable labeling (LCL) problems in the LOCAL model of distributed computing. Since 2016, there has been a substantial body of work examining the possible complexities of LCL problems. For example, it has been established that there are no LCL problems exhibiting deterministic complexities falling between ω(log^∗ n) and o(log n). This line of inquiry has yielded a wealth of algorithmic techniques and insights that are useful for algorithm designers. While the complexity landscape of LCL problems on general graphs, trees, and paths is now well understood, graph classes beyond these three cases remain largely unexplored. Indeed, recent research trends have shifted towards a fine-grained study of special instances within the domains of paths and trees. In this paper, we generalize the line of research on characterizing the complexity landscape of LCL problems to a much broader range of graph classes. We propose a conjecture that characterizes the complexity landscape of LCL problems for an arbitrary class of graphs that is closed under minors, and we prove a part of the conjecture. Some highlights of our findings are as follows. - We establish a simple characterization of the minor-closed graph classes sharing the same deterministic complexity landscape as paths, where O(1), Θ(log^∗ n), and Θ(n) are the only possible complexity classes. - It is natural to conjecture that any minor-closed graph class shares the same complexity landscape as trees if and only if the graph class has bounded treewidth and unbounded pathwidth. We prove the 'only if' part of the conjecture. - For the class of graphs with pathwidth at most k, we show the existence of LCL problems with randomized and deterministic complexities Θ(n), Θ(n^{1/2}), Θ(n^{1/3}), …, Θ(n^{1/k}) and the non-existence of LCL problems whose deterministic complexity is between ω(log^∗ n) and o(n^{1/k}). Consequently, in addition to the well-known complexity landscapes for paths, trees, and general grap

Authors: Yi-Jun Chang, Chang, Yi-Jun

Index Terms: Distributed graph algorithms, locality, InProceedings, Text, doc-type:ResearchArticle, publishedVersion

URL: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.26
Is Part Of LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Deterministic Distributed DFS via Cycle Separators in Planar Graphs

Authors: Jauregui, Benjamin, Montealegre, Pedro, Rapaport, Ivan

Source: Proceedings of the ACM Symposium on Principles of Distributed Computing. :268-277

Availability: http://dl.acm.org/doi/10.1145/3732772.3733558

About randomised distributed graph colouring and graph partition algorithms

Authors: Métivier, Y. ^⁎, Robson, J.M., Saheb-Djahromi, N., Zemmari, A.

Source: In Information and Computation 2010 208(11):1296-1304

New Hardness Results for the LOCAL Model via a Simple Self-Reduction

Authors: Balliu, Alkida, Casagrande, Filippo, d'Amore, Francesco, Olivetti, Dennis

Source: ACM Symposium on Principles of Distributed Computing. :301-310

Availability: http://dl.acm.org/doi/10.1145/3796701.3815938

Node and edge averaged complexities of local graph problems

Additional Titles: Abstract We continue the recently started line of work on the distributed node-averaged complexity of distributed graph algorithms. The node-averaged complexity of a distributed algorithm running on a graph $$G=(V,E)$$ G = ( V , E ) is the average over the times at which the nodes V of G finish their computation and commit to their outputs. We study the node-averaged complexity for some of the central distributed symmetry breaking problems and provide the following results (among others). As our main result, we show that the randomized node-averaged complexity of computing a maximal independent set (MIS) in n-node graphs of maximum degree $$\Delta $$ Δ is at least $$\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\big )$$ Ω ( min { log Δ log log Δ , log n log log n } ) . This bound is obtained by a novel adaptation of the well-known lower bound by Kuhn, Moscibroda, and Wattenhofer [JACM’16]. As a side result, we obtain that the worst-case randomized round complexity for computing an MIS in trees is also $$\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\big )$$ Ω ( min { log Δ log log Δ , log n log log n } ) —this essentially answers open problem 11.15 in the book by Barenboim and Elkin and resolves the complexity of MIS on trees up to an $$O(\sqrt{\log \log n})$$ O ( log log n ) factor. We also show that, perhaps surprisingly, a minimal relaxation of MIS, which is the same as (2, 1)-ruling set, to the (2, 2)-ruling set problem drops the randomized node-averaged complexity to O(1). For maximal matching, we show that while the randomized node-averaged complexity is $$\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\big )$$ Ω ( min { log Δ log log Δ , log n log log n } ) , the randomized edge-averaged complexity is O(1). Further, we show that the deterministic edge-averaged complexity of maximal matchin

Authors: Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science, Balliu, Alkida, Ghaffari, Mohsen, Kuhn, Fabian, Olivetti, Dennis

Source: Springer Berlin Heidelberg

Index Terms: Article, http://purl.org/eprint/type/JournalArticle

URL: https://hdl.handle.net/1721.1/151070
https://doi.org/10.1007/s00446-023-00453-1
https://doi.org/10.1007/s00446-023-00453-1

Byzantine Connectivity Testing in the Congested Clique

Additional Titles: We initiate the study of distributed graph algorithms under the presence of Byzantine nodes. We consider the fundamental problem of testing the connectivity of a graph in the congested clique model in a Byzantine setting. We are given a n-vertex (arbitrary) graph G embedded in a n-node congested clique where an arbitrary subset of B nodes of the clique of size up to (1/3-ε)n (for any arbitrary small constant ε > 0) can be Byzantine. We consider the full information model where Byzantine nodes can behave arbitrarily, collude with each other, and have unlimited computational power and full knowledge of the states and actions of the honest nodes, including random choices made up to the current round. Our main result is an efficient randomized distributed algorithm that is able to correctly distinguish between two contrasting cases: (1) the graph G⧵ B (i.e., the graph induced by the removal of the vertices assigned to the Byzantine nodes in the clique) is connected or (2) the graph G is far from connected, i.e., it has at least 2|B|+1 connected components. Our algorithm runs in O(polylog n) rounds in the congested clique model and guarantees that all honest nodes will decide on the correct case with high probability. Since Byzantine nodes can lie about the vertices assigned to them, we show that this is essentially the best possible that can be done by any algorithm. Our result can be viewed also in the spirit of property testing, where our algorithm is able to distinguish between two contrasting cases while giving no guarantees if the graph falls in the grey area (i.e., neither of the cases occur). Our work is a step towards robust and secure distributed graph computation that can output meaningful results even in the presence of a large number of faulty or malicious nodes.

Authors: John Augustine and Anisur Rahaman Molla and Gopal Pandurangan and Yadu Vasudev, Augustine, John, Molla, Anisur Rahaman, Pandurangan, Gopal, Vasudev, Yadu

Index Terms: Byzantine protocols, distributed graph algorithms, congested clique, graph connectivity, fault-tolerant computation, randomized algorithms, InProceedings, Text, doc-type:ResearchArticle, publishedVersion

URL: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.7
Is Part Of LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

Additional Titles: We provide CONGEST model algorithms for approximating the minimum weighted vertex cover and the maximum weighted matching problem. For bipartite graphs, we show that a (1+ε)-approximate weighted vertex cover can be computed deterministically in poly((log n)/ε) rounds. This generalizes a corresponding result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS '20]. Moreover, we show that in general weighted graph families that are closed under taking subgraphs and in which we can compute an independent set of weight at least λ⋅ w(V) (where w(V) denotes the total weight of all nodes) in polylogarithmic time in the CONGEST model, one can compute a (2-2λ +ε)-approximate weighted vertex cover in poly((log n)/ε) rounds in the CONGEST model. Our result in particular implies that in graphs of arboricity a, one can compute a (2-1/a+ε)-approximate weighted vertex cover problem in poly((log n)/ε) rounds in the CONGEST model. For maximum weighted matchings, we show that a (1-ε)-approximate solution can be computed deterministically in time 2^{O(1/ε)}⋅ polylog n in the CONGEST model. We also provide a randomized algorithm that with arbitrarily good constant probability succeeds in computing a (1-ε)-approximate weighted matching in time 2^{O(1/ε)}⋅ polylog(Δ W)⋅ log^* n, where W denotes the ratio between the largest and the smallest edge weight. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie; SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17], who gave 2^{O(1/ε)}⋅ log n and 2^{O(1/ε)}⋅ (logΔ)/(log logΔ)-round randomized approximations for the unweighted matching problem. Finally, we show that even in the LOCAL model and in bipartite graphs of degree ≤ 3, if ε < ε₀ for some constant ε₀ > 0, then computing a (1+ε)-approximation for the unweighted minimum vertex cover problem requires Ω((log n)/ε) rounds. This generalizes a result of [Göös, Suomela; DISC '12], who showed that computing a (1+ε₀)-approximation in such graphs requir

Authors: Salwa Faour and Marc Fuchs and Fabian Kuhn, Faour, Salwa, Fuchs, Marc, Kuhn, Fabian

Index Terms: distributed graph algorithms, minimum weighted vertex cover, maximum weighted matching, distributed optimization, CONGEST model, InProceedings, Text, doc-type:ResearchArticle, publishedVersion

URL: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.17
Is Part Of LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)

Approximating Bipartite Minimum Vertex Cover in the CONGEST Model

Additional Titles: We give efficient distributed algorithms for the minimum vertex cover problem in bipartite graphs in the CONGEST model. From Kőnig’s theorem, it is well known that in bipartite graphs the size of a minimum vertex cover is equal to the size of a maximum matching. We first show that together with an existing O(nlog n)-round algorithm for computing a maximum matching, the constructive proof of Kőnig’s theorem directly leads to a deterministic O(nlog n)-round CONGEST algorithm for computing a minimum vertex cover. We then show that by adapting the construction, we can also convert an approximate maximum matching into an approximate minimum vertex cover. Given a (1-δ)-approximate matching for some δ > 1, we show that a (1+O(δ))-approximate vertex cover can be computed in time O (D+poly((log n)/δ)), where D is the diameter of the graph. When combining with known graph clustering techniques, for any ε ∈ (0,1], this leads to a poly((log n)/ε)-time deterministic and also to a slightly faster and simpler randomized O((log n)/ε³)-round CONGEST algorithm for computing a (1+ε)-approximate vertex cover in bipartite graphs. For constant ε, the randomized time complexity matches the Ω(log n) lower bound for computing a (1+ε)-approximate vertex cover in bipartite graphs even in the LOCAL model. Our results are also in contrast to the situation in general graphs, where it is known that computing an optimal vertex cover requires Ω̃(n²) rounds in the CONGEST model and where it is not even known how to compute any (2-ε)-approximation in time o(n²).

Authors: Salwa Faour and Fabian Kuhn, Faour, Salwa, Kuhn, Fabian

Index Terms: distributed vertex cover, distributed graph algorithms, distributed optimization, bipartite vertex cover, InProceedings, Text, doc-type:ResearchArticle, publishedVersion

URL: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2020.29
Is Part Of LIPIcs, Volume 184, 24th International Conference on Principles of Distributed Systems (OPODIS 2020)

Brief Announcement: 2-Coloring Cycles in One Round

Authors: Flin, Maxime, Raevskaya, Alesya, Stimpert, Ronja, Suomela, Jukka, Yang, Qingxin

Source: ACM Symposium on Principles of Distributed Computing. :44-47

Availability: http://dl.acm.org/doi/10.1145/3796701.3815937

On the Use of Randomness in Local Distributed Graph Algorithms

Authors: Ghaffari, Mohsen, Kuhn, Fabian

Source: Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing. :290-299

Availability: http://dl.acm.org/doi/10.1145/3293611.3331610

Fast Distributed Algorithms for Connectivity and MST in Large Graphs

Authors: Pandurangan, Gopal, Robinson, Peter, Scquizzato, Michele

Source: Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures. :429-438

Availability: http://dl.acm.org/doi/10.1145/2935764.2935785

Fast distributed algorithms for connectivity and MST in Large Graphs

Additional Titles: Motivated by the increasing need to understand the algorithmic foundations of distributed large-scale graph computations, we study a number of fundamental graph problems in amessage-passing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n > k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-To-point, and the goal is to minimize the number of communication rounds of the computation. Our main result is an (almost) optimal distributed randomized algorithm for graph connectivity. Our algorithm runs in Õ (n/k2) rounds ( Õ notation hides a polylog(n) factor and an additive polylog(n) term). This improves over the best previously known bound of Õ (n/k) [Klauck et al., SODA 2015] and is optimal (up to a polylogarithmic factor) in light of an existing lower bound of ω(n/k2). Our improved algorithm uses a bunch of techniques, including linear graph sketching, that prove useful in the design of efficient distributed graph algorithms. Using the connectivity algorithm as a building block, we then present fast randomized algorithms for computing minimum spanning trees, (approximate) min-cuts, and for many graph verification problems. All these algorithms take Õ (n/k2) rounds and are optimal up to polylogarithmic factors. We also show an almost matching lower bound of ω (n/k2) rounds for many graph verification problems by leveraging lower bounds in random-partition communication complexity.

Authors: Pandurangan, G., Robinson, P., Scquizzato, Michele

Index Terms: Distributed graph algorithms, Graph connectivity, Graph sketching, Minimum spanning trees, Distributed computer systems, Algorithmic foundations, Communication complexity, Connectivity algorithms, Poly-logarithmic factors, Trees (mathematics), Computer Sciences, Datavetenskap (datalogi), Article in journal, info:eu-repo/semantics/article, text

URL: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-314288
2018, 5:1, s. 1-22
ACM Transactions on Parallel Computing, 2329-4949, 2018, 5:1, s. 1-22

Decentralized Distributed Graph Coloring: Cluster Graphs

Authors: Flin, Maxime, Halldórsson, Magnús M., Nolin, Alexandre

Source: Proceedings of the ACM Symposium on Principles of Distributed Computing. :394-405

Availability: http://dl.acm.org/doi/10.1145/3732772.3733549

A Simple Deterministic Distributed MST Algorithm with Near-Optimal Time and Message Complexities

Authors: Elkin, M.

Source: Journal of the ACM (JACM). 67(2):1-15

Availability: http://dl.acm.org/doi/10.1145/3380546

Distributed Graph Realizations

Authors: Augustine, J., Choudhary, K., Cohen, A., Peleg, D., Sivasubramaniam, S., Sourav, S.

Source: IEEE Transactions on Parallel and Distributed Systems IEEE Trans. Parallel Distrib. Syst. Parallel and Distributed Systems, IEEE Transactions on. 33(6):1321-1337 Jun, 2022