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**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

We study the fine-grained complexity of listing all 4-cycles in a graph on n nodes, m edges, and t such 4-cycles. The main result is an Õ(min(n²,m^{4/3})+t) upper bound, which is best-possible up to log factors unless the long-standing O(min(n²,m^{4/3})) upper bound for detecting a 4-cycle can be broken. Moreover, it almost-matches recent 3-SUM-based lower bounds for the problem by Abboud, Bringmann, and Fischer (STOC 2023) and independently by Jin and Xu (STOC 2023). Notably, our result separates 4-cycle listing from the closely related triangle listing for which higher conditional lower bounds exist, and rule out such a "detection plus t" bound. We also show by simple arguments that our bound cannot be extended to mild generalizations of the problem such as reporting all pairs of nodes that participate in a 4-cycle.
[Independent work: Jin and Xu [Ce Jin and Yinzhan Xu, 2023] also present an algorithm with the same time bound.]

Amir Abboud, Seri Khoury, Oree Leibowitz, and Ron Safier. Listing 4-Cycles. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abboud_et_al:LIPIcs.FSTTCS.2023.25, author = {Abboud, Amir and Khoury, Seri and Leibowitz, Oree and Safier, Ron}, title = {{Listing 4-Cycles}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {25:1--25:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.25}, URN = {urn:nbn:de:0030-drops-193985}, doi = {10.4230/LIPIcs.FSTTCS.2023.25}, annote = {Keywords: Graph algorithms, cycles listing, subgraph detection, fine-grained complexity} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

We study the problem of approximating the diameter D of an unweighted and undirected n-node graph in the congest model. Through a connection to extremal combinatorics, we show that a (6/11 + ε)-approximation requires Ω(n^{1/6}/log n) rounds, a (4/7 + ε)-approximation requires Ω(n^{1/4}/log n) rounds, and a (3/5 + ε)-approximation requires Ω(n^{1/3}/log n) rounds. These lower bounds are robust in the sense that they hold even against algorithms that are allowed to return an additional small additive error. Prior to our work, only lower bounds for (2/3 + ε)-approximation were known [Frischknecht et al. SODA 2012, Abboud et al. DISC 2016].
Furthermore, we prove that distinguishing graphs of diameter 3 from graphs of diameter 5 requires Ω(n/log n) rounds. This stands in sharp contrast to previous work: while there is an algorithm that returns an estimate ⌊ 2/3D ⌋ ≤ D̃ ≤ D in Õ(√n+D) rounds [Holzer et al. DISC 2014], our lower bound implies that any algorithm for returning an estimate 2/3D ≤ D̃ ≤ D requires ̃Ω(n) rounds.

Ofer Grossman, Seri Khoury, and Ami Paz. Improved Hardness of Approximation of Diameter in the CONGEST Model. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{grossman_et_al:LIPIcs.DISC.2020.19, author = {Grossman, Ofer and Khoury, Seri and Paz, Ami}, title = {{Improved Hardness of Approximation of Diameter in the CONGEST Model}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {19:1--19:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.19}, URN = {urn:nbn:de:0030-drops-130972}, doi = {10.4230/LIPIcs.DISC.2020.19}, annote = {Keywords: Distributed graph algorithms, Approximation algorithms, Lower bounds} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

We present improved results for approximating maximum-weight independent set (MaxIS) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let n and Δ be the number of nodes and maximum degree, respectively, and let MIS(n,Δ) be the running time of finding a maximal independent set (MIS) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation for MaxIS in O(MIS(n,Δ)log W) rounds, where W is the maximum weight of a node in the graph, which can be as large as poly (n). Whether their algorithm is deterministic or randomized that succeeds with high probability depends on the MIS algorithm that is used as a black-box. Our results:
1) A deterministic O(MIS(n,Δ)/ε)-round algorithm that finds a (1+ε)Δ-approximation for MaxIS in the CONGEST model.
2) A randomized (poly(log log n)/ε)-round algorithm that finds, with high probability, a (1+ε)Δ-approximation for MaxIS in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an exponential speed-up in the running time over the previous best known result.
3) A randomized O(log n⋅ poly(log log n)/ε)-round algorithm that finds, with high probability, a 8(1+ε)α-approximation for MaxIS in the CONGEST model, where α is the arboricity of the graph. For graphs of arboricity α < Δ/(8(1+ε)), this result improves upon the previous best known result in both the approximation factor and the running time.
One may wonder whether it is possible to approximate MaxIS with high probability in fewer than poly(log log n) rounds. Interestingly, a folklore randomized ranking algorithm by Boppana implies a single round algorithm that gives an expected Δ-approximation in the CONGEST model. However, it is unclear how to convert this algorithm to one that succeeds with high probability without sacrificing a large number of rounds. For unweighted graphs of maximum degree Δ ≤ n/log n, we show a new analysis of the randomized ranking algorithm, which we combine with the local-ratio technique, to provide a O(1/ε)-round algorithm in the CONGEST model that, with high probability, finds an independent set of size at least n/((1+ε)(Δ+1)). This result cannot be extended to very high degree graphs, as we show a lower bound of Ω(log^*n) rounds for any randomized algorithm that with probability at least 1-1/log n finds an independent set of size Ω(n/Δ). This lower bound holds even for the LOCAL model. The hard instances that we use to prove our lower bound are graphs of maximum degree Δ = Ω(n/log^*n).

Ken-ichi Kawarabayashi, Seri Khoury, Aaron Schild, and Gregory Schwartzman. Improved Distributed Approximations for Maximum Independent Set. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kawarabayashi_et_al:LIPIcs.DISC.2020.35, author = {Kawarabayashi, Ken-ichi and Khoury, Seri and Schild, Aaron and Schwartzman, Gregory}, title = {{Improved Distributed Approximations for Maximum Independent Set}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {35:1--35:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.35}, URN = {urn:nbn:de:0030-drops-131135}, doi = {10.4230/LIPIcs.DISC.2020.35}, annote = {Keywords: Distributed graph algorithms, Approximation algorithms, Lower bounds} }

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

We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question.
Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires a near-quadratic number of rounds in the CONGEST model, as well as any algorithm for computing the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of rounds.
Finally, we address the problem of computing an exact solution to weighted all-pairs-shortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple linear lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a sub-linear complexity. We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question.

Keren Censor-Hillel, Seri Khoury, and Ami Paz. Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{censorhillel_et_al:LIPIcs.DISC.2017.10, author = {Censor-Hillel, Keren and Khoury, Seri and Paz, Ami}, title = {{Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {10:1--10:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.10}, URN = {urn:nbn:de:0030-drops-79969}, doi = {10.4230/LIPIcs.DISC.2017.10}, annote = {Keywords: CONGEST, Lower Bounds, Minimum Vertex Cover, Chromatic Number, Weighted APSP} }

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