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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

The message complexity of a distributed algorithm is the total number of messages sent by all nodes over the course of the algorithm. This paper studies the message complexity of distributed algorithms for fundamental graph optimization problems. We focus on four classical graph optimization problems: Maximum Matching (MaxM), Minimum Vertex Cover (MVC), Minimum Dominating Set (MDS), and Maximum Independent Set (MaxIS). In the sequential setting, these problems are representative of a wide spectrum of hardness of approximation. While there has been some progress in understanding the round complexity of distributed algorithms (for both exact and approximate versions) for these problems, much less is known about their message complexity and its relation with the quality of approximation. We almost fully quantify the message complexity of distributed graph optimization by showing the following results:
1) Cubic regime: Our first main contribution is showing essentially cubic, i.e., Ω̃(n³) lower bounds (where n is the number of nodes in the graph) on the message complexity of distributed exact computation of Minimum Vertex Cover (MVC), Minimum Dominating Set (MDS), and Maximum Independent Set (MaxIS). Our lower bounds apply to any distributed algorithm that runs in polynomial number of rounds (a mild and necessary restriction). Our result is significant since, to the best of our knowledge, this are the first ω(m) (where m is the number of edges in the graph) message lower bound known for distributed computation of such classical graph optimization problems. Our bounds are essentially tight, as all these problems can be solved trivially using O(n³) messages in polynomial rounds. All these bounds hold in the standard CONGEST model of distributed computation in which messages are of O(log n) size.
2) Quadratic regime: In contrast, we show that if we allow approximate computation then Θ̃(n²) messages are both necessary and sufficient. Specifically, we show that Ω̃(n²) messages are required for constant-factor approximation algorithms for all four problems. For MaxM and MVC, these bounds hold for any constant-factor approximation, whereas for MDS and MaxIS they hold for any approximation factor better than some specific constants. These lower bounds hold even in the LOCAL model (in which messages can be arbitrarily large) and they even apply to algorithms that take arbitrarily many rounds. We show that our lower bounds are essentially tight, by showing that if we allow approximation to within an arbitrarily small constant factor, then all these problems can be solved using Õ(n²) messages even in the CONGEST model.
3) Linear regime: We complement the above lower bounds by showing distributed algorithms with Õ(n) message complexity that run in polylogarithmic rounds and give constant-factor approximations for all four problems on random graphs. These results imply that almost linear (in n) message complexity is achievable on almost all (connected) graphs of every edge density.

Fabien Dufoulon, Shreyas Pai, Gopal Pandurangan, Sriram V. Pemmaraju, and Peter Robinson. The Message Complexity of Distributed Graph Optimization. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 41:1-41:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dufoulon_et_al:LIPIcs.ITCS.2024.41, author = {Dufoulon, Fabien and Pai, Shreyas and Pandurangan, Gopal and Pemmaraju, Sriram V. and Robinson, Peter}, title = {{The Message Complexity of Distributed Graph Optimization}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {41:1--41:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.41}, URN = {urn:nbn:de:0030-drops-195690}, doi = {10.4230/LIPIcs.ITCS.2024.41}, annote = {Keywords: Distributed graph algorithm, message complexity, distributed approximation} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We consider the problem of computing a maximal matching with a distributed algorithm in the presence of batch-dynamic changes to the graph topology. We assume that a graph of n nodes is vertex-partitioned among k players that communicate via message passing. Our goal is to provide an efficient algorithm that quickly updates the matching even if an adversary determines batches of 𝓁 edge insertions or deletions. We first show a lower bound of Ω((𝓁 log k)/(k²log n)) rounds for recomputing a matching assuming an oblivious adversary who is unaware of the initial (random) vertex partition as well as the current state of the players, and a stronger lower bound of Ω(𝓁/(klog n)) rounds against an adaptive adversary, who may choose any balanced (but not necessarily random) vertex partition initially and who knows the current state of the players. We also present a randomized algorithm that has an initialization time of O(n/(k log n)) rounds, while achieving an update time that that is independent of n: In more detail, the update time is O(⌈𝓁/k⌉ log k) against an oblivious adversary, who must fix all updates in advance. If we consider the stronger adaptive adversary, the update time becomes O (⌈𝓁/√k⌉ log k) rounds.

Minming Li, Peter Robinson, and Xianbin Zhu. Dynamic Maximal Matching in Clique Networks. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 73:1-73:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.ITCS.2024.73, author = {Li, Minming and Robinson, Peter and Zhu, Xianbin}, title = {{Dynamic Maximal Matching in Clique Networks}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {73:1--73:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.73}, URN = {urn:nbn:de:0030-drops-196017}, doi = {10.4230/LIPIcs.ITCS.2024.73}, annote = {Keywords: distributed graph algorithm, dynamic network, maximal matching, randomized algorithm, lower bound} }

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**Published in:** LIPIcs, Volume 286, 27th International Conference on Principles of Distributed Systems (OPODIS 2023)

We consider the message complexity of verifying whether a given subgraph of the communication network forms a tree with specific properties both in the KT_ρ (nodes know their ρ-hop neighborhood, including node ids) and the KT₀ (nodes do not have this knowledge) models. We develop a rather general framework that helps in establishing tight lower bounds for various tree verification problems. We also consider two different verification requirements: namely that every node detects in the case the input is incorrect, as well as the requirement that at least one node detects. The results are stronger than previous ones in the sense that we assume that each node knows the number n of nodes in the graph (in some cases) or an α approximation of n (in other cases). For spanning tree verification, we show that the message complexity inherently depends on the quality of the given approximation of n: We show a tight lower bound of Ω(n²) for the case α ≥ √2 and a much better upper bound (i.e., O(n log n)) when nodes are given a tighter approximation. On the other hand, our framework also yields an Ω(n²) lower bound on the message complexity of verifying a minimum spanning tree (MST), which reveals a polynomial separation between ST verification and MST verification. This result holds for randomized algorithms with perfect knowledge of the network size, and even when just one node detects illegal inputs, thus improving over the work of Kor, Korman, and Peleg (2013). For verifying a d-approximate BFS tree, we show that the same lower bound holds even if nodes know n exactly, however, the lower bounds is sensitive to d, which is the stretch parameter. First, under the KT₀ assumption, we show a tight message complexity lower bound of Ω(n²) in the LOCAL model, when d ≤ n/(2+Ω(1)). For the KT_ρ assumption, we obtain an upper bound on the message complexity of O(nlog n) in the CONGEST model, when d ≥ (n-1)/max{2,ρ+1}, and use a novel charging argument to show that Ω((1/ρ)(n/ρ)^{1+c/ρ}) messages are required even in the LOCAL model for comparison-based algorithms. For the well-studied special case of KT₁, we obtain a tight lower bound of Ω(n²).

Shay Kutten, Peter Robinson, and Ming Ming Tan. Tight Bounds on the Message Complexity of Distributed Tree Verification. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 26:1-26:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kutten_et_al:LIPIcs.OPODIS.2023.26, author = {Kutten, Shay and Robinson, Peter and Tan, Ming Ming}, title = {{Tight Bounds on the Message Complexity of Distributed Tree Verification}}, booktitle = {27th International Conference on Principles of Distributed Systems (OPODIS 2023)}, pages = {26:1--26:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-308-9}, ISSN = {1868-8969}, year = {2024}, volume = {286}, editor = {Bessani, Alysson and D\'{e}fago, Xavier and Nakamura, Junya and Wada, Koichi and Yamauchi, Yukiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2023.26}, URN = {urn:nbn:de:0030-drops-195163}, doi = {10.4230/LIPIcs.OPODIS.2023.26}, annote = {Keywords: Distributed Graph Algorithm, Lower Bound} }

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**Published in:** LIPIcs, Volume 281, 37th International Symposium on Distributed Computing (DISC 2023)

We investigate graph problems in the distributed sketching model, where each node sends a single message to a referee who computes the output. We define a class of graphs and give a framework for proving lower bounds for certain embeddable problems, which leads to several new results: First, we present a tight lower bound of Ω(n) bits for the message size of computing a breadth-first search (BFS) tree. For locally-checkable labeling (LCL) problems, we show that verifying whether a given vertex labeling forms a weak 2-coloring requires messages of Ω(n^{1/3}log^{2/3}n) bits, and the same lower bound holds for verifying whether a subset of nodes forms a maximal independent set. We also prove that computing a k-edge connected spanning subgraph (k-ECSS) requires messages of size at least Ω(klog²(n/k)), which is tight up to a logarithmic factor. To show these results, we define a simultaneous multiparty (SMP) model of communication complexity, where the players obtain certain subgraphs as their input, and develop a generic embedding argument that allows us to prove lower bounds for the graph sketching model by using reductions from the SMP model. We point out that these results also extend to single-round algorithms in the broadcast congested clique.
We also (nearly) settle the space complexity of the k-ECSS problem in the streaming model by extending the work of Kapralov, Nelson, Pachoki, Wang, and Woodruff (FOCS 2017): We prove a communication complexity lower bound for a direct sum variant of the UR^⊂_k problem and show that this implies Ω(k nlog²(n/k)) bits of memory for computing a k-ECSS. This is known to be optimal up to a logarithmic factor.

Peter Robinson. Distributed Sketching Lower Bounds for k-Edge Connected Spanning Subgraphs, BFS Trees, and LCL Problems. In 37th International Symposium on Distributed Computing (DISC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 281, pp. 32:1-32:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{robinson:LIPIcs.DISC.2023.32, author = {Robinson, Peter}, title = {{Distributed Sketching Lower Bounds for k-Edge Connected Spanning Subgraphs, BFS Trees, and LCL Problems}}, booktitle = {37th International Symposium on Distributed Computing (DISC 2023)}, pages = {32:1--32:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-301-0}, ISSN = {1868-8969}, year = {2023}, volume = {281}, editor = {Oshman, Rotem}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2023.32}, URN = {urn:nbn:de:0030-drops-191589}, doi = {10.4230/LIPIcs.DISC.2023.32}, annote = {Keywords: Distributed graph algorithm, graph sketching, streaming} }

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**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related beta-ruling set problem, in two computational models suited for large-scale graph processing, namely the k-machine model and the graph streaming model. We present a number of results. For MIS in the k-machine model, we improve the O~(m/k^2 + Delta/k)-round upper bound of Klauck et al. (SODA 2015) by presenting an O~(m/k^2)-round algorithm. We also present an Omega~(n/k^2) round lower bound for MIS, the first lower bound for a symmetry breaking problem in the k-machine model. For beta-ruling sets, we use hierarchical sampling to obtain more efficient algorithms in the k-machine model and also in the graph streaming model. More specifically, we obtain a k-machine algorithm that runs in O~(beta n Delta^{1/beta}/k^2) rounds and, by using a similar hierarchical sampling technique, we obtain one-pass algorithms for both insertion-only and insertion-deletion streams that use O(beta * n^{1+1/2^{beta-1}}) space. The latter result establishes a clear separation between MIS, which is known to require Omega(n^2) space (Cormode et al., ICALP 2019), and beta-ruling sets, even for beta = 2. Finally, we present an even faster 2-ruling set algorithm in the k-machine model, one that runs in O~(n/k^{2-epsilon} + k^{1-epsilon}) rounds for any epsilon, 0 <=epsilon <=1. For a wide range of values of k this round complexity simplifies to O~(n/k^2) rounds, which we conjecture is optimal.
Our results use a variety of techniques. For our upper bounds, we prove and use simulation theorems for beeping algorithms, hierarchical sampling, and L_0-sampling, whereas for our lower bounds we use information-theoretic arguments and reductions to 2-party communication complexity problems.

Christian Konrad, Sriram V. Pemmaraju, Talal Riaz, and Peter Robinson. The Complexity of Symmetry Breaking in Massive Graphs. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{konrad_et_al:LIPIcs.DISC.2019.26, author = {Konrad, Christian and Pemmaraju, Sriram V. and Riaz, Talal and Robinson, Peter}, title = {{The Complexity of Symmetry Breaking in Massive Graphs}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {26:1--26:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.26}, URN = {urn:nbn:de:0030-drops-113337}, doi = {10.4230/LIPIcs.DISC.2019.26}, annote = {Keywords: communication complexity, information theory, k-machine model, maximal independent set, ruling set, streaming algorithms} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

In this paper, we study fault-tolerant distributed consensus in wireless systems. In more detail, we produce two new randomized algorithms that solve this problem in the abstract MAC layer model, which captures the basic interface and communication guarantees provided by most wireless MAC layers. Our algorithms work for any number of failures, require no advance knowledge of the network participants or network size, and guarantee termination with high probability after a number of broadcasts that are polynomial in the network size. Our first algorithm satisfies the standard agreement property, while our second trades a faster termination guarantee in exchange for a looser agreement property in which most nodes agree on the same value. These are the first known fault-tolerant consensus algorithms for this model. In addition to our main upper bound results, we explore the gap between the abstract MAC layer and the standard asynchronous message passing model by proving fault-tolerant consensus is impossible in the latter in the absence of information regarding the network participants, even if we assume no faults, allow randomized solutions, and provide the algorithm a constant-factor approximation of the network size.

Calvin Newport and Peter Robinson. Fault-Tolerant Consensus with an Abstract MAC Layer. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{newport_et_al:LIPIcs.DISC.2018.38, author = {Newport, Calvin and Robinson, Peter}, title = {{Fault-Tolerant Consensus with an Abstract MAC Layer}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {38:1--38:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.38}, URN = {urn:nbn:de:0030-drops-98277}, doi = {10.4230/LIPIcs.DISC.2018.38}, annote = {Keywords: abstract MAC layer, wireless networks, consensus, fault tolerance} }

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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree Delta is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for moderately small Delta (i.e., for Delta = Omega(log n) and Delta = o(n)). Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Theta(log n) time complexity barrier and the Theta(m) message complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems?
This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A beta-ruling set is an independent set such that every node in the graph is at most beta hops from a node in the independent set. We present the following results:
- Time Complexity: We show that we can break the O(log n) "barrier" for 2- and 3-ruling sets. We compute 3-ruling sets in O(log n/log log n) rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O(log Delta (log n)^(1/2 + epsilon) + log n/log log n) rounds for any epsilon > 0, which is o(log n) for a wide range of Delta values (e.g., Delta = 2^(log n)^(1/2-epsilon)). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model.
- Message Complexity: We show an Omega(n^2) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log^2 n) messages and runs in O(Delta log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor).

Shreyas Pai, Gopal Pandurangan, Sriram V. Pemmaraju, Talal Riaz, and Peter Robinson. Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pai_et_al:LIPIcs.DISC.2017.38, author = {Pai, Shreyas and Pandurangan, Gopal and Pemmaraju, Sriram V. and Riaz, Talal and Robinson, Peter}, title = {{Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {38:1--38:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.38}, URN = {urn:nbn:de:0030-drops-80132}, doi = {10.4230/LIPIcs.DISC.2017.38}, annote = {Keywords: Congest model, Local model, Maximal independent set, Message complexity, Round complexity, Ruling sets, Symmetry breaking} }

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