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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.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 focus on the well-studied problem of distributed overlay network construction. We consider a synchronous gossip-based communication model where in each round a node can send a message of small size to another node whose identifier it knows. The network is assumed to be reconfigurable, i.e., a node can add new connections (edges) to other nodes whose identifier it knows or drop existing connections. Each node initially has only knowledge of its own identifier and the identifiers of its neighbors. The overlay construction problem is, given an arbitrary (connected) graph, to reconfigure it to obtain a bounded-degree expander graph as efficiently as possible. The overlay construction problem is relevant to building real-world peer-to-peer network topologies that have desirable properties such as low diameter, high conductance, robustness to adversarial deletions, etc.
Our main result is that we show that starting from any arbitrary (connected) graph G on n nodes and m edges, we can construct an overlay network that is a constant-degree expander in polylog rounds using only Õ(n) messages. Our time and message bounds are both essentially optimal (up to polylogarithmic factors). Our distributed overlay construction protocol is very lightweight as it uses gossip (each node communicates with only one neighbor in each round) and also scalable as it uses only Õ(n) messages, which is sublinear in m (even when m is moderately dense). To the best of our knowledge, this is the first result that achieves overlay network construction in polylog rounds and o(m) messages. Our protocol uses graph sketches in a novel way to construct an expander overlay that is both time and communication efficient.
A consequence of our overlay construction protocol is that distributed computation can be performed very efficiently in this model. In particular, a wide range of fundamental tasks such as broadcast, leader election, and minimum spanning tree (MST) construction can be accomplished in polylog rounds and Õ(n) message complexity in any graph.

Fabien Dufoulon, Michael Moorman, William K. Moses Jr., and Gopal Pandurangan. Time- and Communication-Efficient Overlay Network Construction via Gossip. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 42:1-42:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dufoulon_et_al:LIPIcs.ITCS.2024.42, author = {Dufoulon, Fabien and Moorman, Michael and Moses Jr., William K. and Pandurangan, Gopal}, title = {{Time- and Communication-Efficient Overlay Network Construction via Gossip}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {42:1--42:23}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.42}, URN = {urn:nbn:de:0030-drops-195700}, doi = {10.4230/LIPIcs.ITCS.2024.42}, annote = {Keywords: Peer-to-Peer Networks, Overlay Construction Protocol, Gossip, Expanders, Sublinear Bounds} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Constructing a shortest path between two network nodes is a fundamental task in distributed computing. This work develops schemes for the construction of shortest paths in randomized beeping networks between a predetermined source node and an arbitrary set of destination nodes. Our first scheme constructs a (single) shortest path to an arbitrary destination in O(D log log n + log³ n) rounds with high probability. Our second scheme constructs multiple shortest paths, one per each destination, in O(D log² n + log³ n) rounds with high probability.
Our schemes are based on a reduction of the above shortest path construction tasks to a decomposition of hypergraphs into bipartite hypergraphs: We develop a beeping procedure that partitions the hyperedge set of a hypergraph H = (V_H, E_H) into k = Θ (log² n) disjoint subsets F₁ ∪ ⋯ ∪ F_k = E_H such that the (sub-)hypergraph (V_H, F_i) is bipartite in the sense that there exists a vertex subset U ⊆ V such that |U ∩ e| = 1 for every e ∈ F_i. This procedure turns out to be instrumental in speeding up shortest path constructions under the beeping model.

Fabien Dufoulon, Yuval Emek, and Ran Gelles. Beeping Shortest Paths via Hypergraph Bipartite Decomposition. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 45:1-45:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dufoulon_et_al:LIPIcs.ITCS.2023.45, author = {Dufoulon, Fabien and Emek, Yuval and Gelles, Ran}, title = {{Beeping Shortest Paths via Hypergraph Bipartite Decomposition}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {45:1--45:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.45}, URN = {urn:nbn:de:0030-drops-175485}, doi = {10.4230/LIPIcs.ITCS.2023.45}, annote = {Keywords: Beeping Networks, Shortest Paths, Hypergraph Bipartite Decomposition} }

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**Published in:** LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

A singularly (near) optimal distributed algorithm is one that is (near) optimal in two criteria, namely, its time and message complexities. For synchronous CONGEST networks, such algorithms are known for fundamental distributed computing problems such as leader election [Kutten et al., JACM 2015] and Minimum Spanning Tree (MST) construction [Pandurangan et al., STOC 2017, Elkin, PODC 2017]. However, it is open whether a singularly (near) optimal bound can be obtained for the MST construction problem in general asynchronous CONGEST networks.
In this paper, we present a randomized distributed MST algorithm that, with high probability, computes an MST in asynchronous CONGEST networks and takes Õ(D^{1+ε} + √n) time and Õ(m) messages, where n is the number of nodes, m the number of edges, D is the diameter of the network, and ε > 0 is an arbitrarily small constant (both time and message bounds hold with high probability). Since Ω̃(D+√n) and Ω(m) are respective time and message lower bounds for distributed MST construction in the standard KT₀ model, our algorithm is message optimal (up to a polylog(n) factor) and almost time optimal (except for a D^ε factor). Our result answers an open question raised in Mashregi and King [DISC 2019] by giving the first known asynchronous MST algorithm that has sublinear time (for all D = O(n^{1-ε})) and uses Õ(m) messages. Using a result of Mashregi and King [DISC 2019], this also yields the first asynchronous MST algorithm that is sublinear in both time and messages in the KT₁ CONGEST model.
A key tool in our algorithm is the construction of a low diameter rooted spanning tree in asynchronous CONGEST that has depth Õ(D^{1+ε}) (for an arbitrarily small constant ε > 0) in Õ(D^{1+ε}) time and Õ(m) messages. To the best of our knowledge, this is the first such construction that is almost singularly optimal in the asynchronous setting. This tree construction may be of independent interest as it can also be used for efficiently performing basic tasks such as verified broadcast and convergecast in asynchronous networks.

Fabien Dufoulon, Shay Kutten, William K. Moses Jr., Gopal Pandurangan, and David Peleg. An Almost Singularly Optimal Asynchronous Distributed MST Algorithm. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dufoulon_et_al:LIPIcs.DISC.2022.19, author = {Dufoulon, Fabien and Kutten, Shay and Moses Jr., William K. and Pandurangan, Gopal and Peleg, David}, title = {{An Almost Singularly Optimal Asynchronous Distributed MST Algorithm}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {19:1--19:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-255-6}, ISSN = {1868-8969}, year = {2022}, volume = {246}, editor = {Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.19}, URN = {urn:nbn:de:0030-drops-172107}, doi = {10.4230/LIPIcs.DISC.2022.19}, annote = {Keywords: Asynchronous networks, Minimum Spanning Tree, Distributed Algorithm, Singularly Optimal} }

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

The beeping model is an extremely restrictive broadcast communication model that relies only on carrier sensing. In this model, we solve the leader election problem with an asymptotically optimal round complexity of O(D + log n), for a network of unknown size n and unknown diameter D (but with unique identifiers). Contrary to the best previously known algorithms in the same setting, the proposed one is deterministic. The techniques we introduce give a new insight as to how local constraints on the exchangeable messages can result in efficient algorithms, when dealing with the beeping model.
Using this deterministic leader election algorithm, we obtain a randomized leader election algorithm for anonymous networks with an asymptotically optimal round complexity of O(D + log n) w.h.p. In previous works this complexity was obtained in expectation only.
Moreover, using deterministic leader election, we obtain efficient algorithms for symmetry-breaking and communication procedures: O(log n) time MIS and 5-coloring for tree networks (which is time-optimal), as well as k-source multi-broadcast for general graphs in O(min(k,log n) * D + k log{(n M)/k}) rounds (for messages in {1,..., M}). This latter result improves on previous solutions when the number of sources k is sublogarithmic (k = o(log n)).

Fabien Dufoulon, Janna Burman, and Joffroy Beauquier. Beeping a Deterministic Time-Optimal Leader Election. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dufoulon_et_al:LIPIcs.DISC.2018.20, author = {Dufoulon, Fabien and Burman, Janna and Beauquier, Joffroy}, title = {{Beeping a Deterministic Time-Optimal Leader Election}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {20:1--20:17}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.20}, URN = {urn:nbn:de:0030-drops-98090}, doi = {10.4230/LIPIcs.DISC.2018.20}, annote = {Keywords: distributed algorithms, leader election, beeping model, time complexity, deterministic algorithms, wireless networks} }

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