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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We study the Online Traveling Salesperson Problem (OLTSP) with predictions. In OLTSP, a sequence of initially unknown requests arrive over time at points (locations) of a metric space. The goal is, starting from a particular point of the metric space (the origin), to serve all these requests while minimizing the total time spent. The server moves with unit speed or is "waiting" (zero speed) at some location. We consider two variants: in the open variant, the goal is achieved when the last request is served. In the closed one, the server additionally has to return to the origin. We adopt a prediction model, introduced for OLTSP on the line [Gouleakis et al., 2023], in which the predictions correspond to the locations of the requests and extend it to more general metric spaces.
We first propose an oracle-based algorithmic framework, inspired by previous work [Bampis et al., 2023]. This framework allows us to design online algorithms for general metric spaces that provide competitive ratio guarantees which, given perfect predictions, beat the best possible classical guarantee (consistency). Moreover, they degrade gracefully along with the increase in error (smoothness), but always within a constant factor of the best known competitive ratio in the classical case (robustness).
Having reduced the problem to designing suitable efficient oracles, we describe how to achieve this for general metric spaces as well as specific metric spaces (rings, trees and flowers), the resulting algorithms being tractable in the latter case. The consistency guarantees of our algorithms are tight in almost all cases, and their smoothness guarantees only suffer a linear dependency on the error, which we show is necessary. Finally, we provide robustness guarantees improving previous results.

Evripidis Bampis, Bruno Escoffier, Themis Gouleakis, Niklas Hahn, Kostas Lakis, Golnoosh Shahkarami, and Michalis Xefteris. Learning-Augmented Online TSP on Rings, Trees, Flowers and (Almost) Everywhere Else. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bampis_et_al:LIPIcs.ESA.2023.12, author = {Bampis, Evripidis and Escoffier, Bruno and Gouleakis, Themis and Hahn, Niklas and Lakis, Kostas and Shahkarami, Golnoosh and Xefteris, Michalis}, title = {{Learning-Augmented Online TSP on Rings, Trees, Flowers and (Almost) Everywhere Else}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.12}, URN = {urn:nbn:de:0030-drops-186659}, doi = {10.4230/LIPIcs.ESA.2023.12}, annote = {Keywords: TSP, Online algorithms, Learning-augmented algorithms, Algorithms with predictions, Competitive analysis} }

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

In this paper, we refine the (almost) existentially optimal distributed Laplacian solver recently developed by Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS `21) into an (almost) universally optimal distributed Laplacian solver.
Specifically, when the topology is known (i.e., the Supported-CONGEST model), we show that any Laplacian system on an n-node graph with shortcut quality SQ(G) can be solved after n^{o(1)} SQ(G) log(1/ε) rounds, where ε is the required accuracy. This almost matches our lower bound that guarantees that any correct algorithm on G requires Ω̃(SQ(G)) rounds, even for a crude solution with ε ≤ 1/2. Several important implications hold in the unknown-topology (i.e., standard CONGEST) case: for excluded-minor graphs we get an almost universally optimal algorithm that terminates in D ⋅ n^{o(1)} log(1/ε) rounds, where D is the hop-diameter of the network; as well as n^{o(1)} log (1/ε)-round algorithms for the case of SQ(G) ≤ n^{o(1)}, which holds for most networks of interest. Conditioned on improvements in state-of-the-art constructions of low-congestion shortcuts, the CONGEST results will match the Supported-CONGEST ones.
Moreover, following a recent line of work in distributed algorithms, we consider a hybrid communication model which enhances CONGEST with limited global power in the form of the node-capacitated clique (NCC) model. In this model, we show the existence of a Laplacian solver with round complexity n^{o(1)} log(1/ε).
The unifying thread of these results, and our main technical contribution, is the study of a novel ρ-congested generalization of the standard part-wise aggregation problem. We develop near-optimal algorithms for this primitive in the Supported-CONGEST model, almost-optimal algorithms in (standard) CONGEST (with the additional overhead due to standard barriers), as well as a simple algorithm for bounded-treewidth graphs with a quadratic dependence on the congestion ρ. This primitive can be readily used to accelerate the Laplacian solver of Forster, Goranci, Liu, Peng, Sun, and Ye, and we believe it will find further independent applications in the future.

Ioannis Anagnostides, Christoph Lenzen, Bernhard Haeupler, Goran Zuzic, and Themis Gouleakis. Almost Universally Optimal Distributed Laplacian Solvers via Low-Congestion Shortcuts. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{anagnostides_et_al:LIPIcs.DISC.2022.6, author = {Anagnostides, Ioannis and Lenzen, Christoph and Haeupler, Bernhard and Zuzic, Goran and Gouleakis, Themis}, title = {{Almost Universally Optimal Distributed Laplacian Solvers via Low-Congestion Shortcuts}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {6:1--6:20}, 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.2022.6}, URN = {urn:nbn:de:0030-drops-171978}, doi = {10.4230/LIPIcs.DISC.2022.6}, annote = {Keywords: Distributed algorithms, Laplacian solvers, low-congestion shortcuts} }

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**Published in:** LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

The HYBRID model was recently introduced by Augustine et al. [John Augustine et al., 2020] in order to characterize from an algorithmic standpoint the capabilities of networks which combine multiple communication modes. Concretely, it is assumed that the standard LOCAL model of distributed computing is enhanced with the feature of all-to-all communication, but with very limited bandwidth, captured by the node-capacitated clique (NCC). In this work we provide several new insights on the power of hybrid networks for fundamental problems in distributed algorithms.
First, we present a deterministic algorithm which solves any problem on a sparse n-node graph in 𝒪̃(√n) rounds of HYBRID, where the notation 𝒪̃(⋅) suppresses polylogarithmic factors of n. We combine this primitive with several sparsification techniques to obtain efficient distributed algorithms for general graphs. Most notably, for the all-pairs shortest paths problem we give deterministic (1 + ε)- and log n/log log n-approximate algorithms for unweighted and weighted graphs respectively with round complexity 𝒪̃(√n) in HYBRID, closely matching the performance of the state of the art randomized algorithm of Kuhn and Schneider [Kuhn and Schneider, 2020]. Moreover, we make a connection with the Ghaffari-Haeupler framework of low-congestion shortcuts [Mohsen Ghaffari and Bernhard Haeupler, 2016], leading - among others - to a (1 + ε)-approximate algorithm for Min-Cut after 𝒪(polylog (n)) rounds, with high probability, even if we restrict local edges to transfer 𝒪(log n) bits per round. Finally, we prove via a reduction from the set disjointness problem that Ω̃(n^{1/3}) rounds are required to determine the radius of an unweighted graph, as well as a (3/2 - ε)-approximation for weighted graphs. As a byproduct, we show an Ω̃(n) round-complexity lower bound for computing a (4/3 - ε)-approximation of the radius in the broadcast variant of the congested clique, even for unweighted graphs.

Ioannis Anagnostides and Themis Gouleakis. Deterministic Distributed Algorithms and Lower Bounds in the Hybrid Model. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{anagnostides_et_al:LIPIcs.DISC.2021.5, author = {Anagnostides, Ioannis and Gouleakis, Themis}, title = {{Deterministic Distributed Algorithms and Lower Bounds in the Hybrid Model}}, booktitle = {35th International Symposium on Distributed Computing (DISC 2021)}, pages = {5:1--5:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-210-5}, ISSN = {1868-8969}, year = {2021}, volume = {209}, editor = {Gilbert, Seth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.5}, URN = {urn:nbn:de:0030-drops-148077}, doi = {10.4230/LIPIcs.DISC.2021.5}, annote = {Keywords: Distributed Computing, Hybrid Model, Sparse Graphs, Deterministic Algorithms, All-Pairs Shortest Paths, Minimum Cut, Radius} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We study the problem of testing identity against a given distribution with a focus on the high confidence regime. More precisely, given samples from an unknown distribution p over n elements, an explicitly given distribution q, and parameters 0< epsilon, delta < 1, we wish to distinguish, with probability at least 1-delta, whether the distributions are identical versus epsilon-far in total variation distance. Most prior work focused on the case that delta = Omega(1), for which the sample complexity of identity testing is known to be Theta(sqrt{n}/epsilon^2). Given such an algorithm, one can achieve arbitrarily small values of delta via black-box amplification, which multiplies the required number of samples by Theta(log(1/delta)).
We show that black-box amplification is suboptimal for any delta = o(1), and give a new identity tester that achieves the optimal sample complexity. Our new upper and lower bounds show that the optimal sample complexity of identity testing is Theta((1/epsilon^2) (sqrt{n log(1/delta)} + log(1/delta))) for any n, epsilon, and delta. For the special case of uniformity testing, where the given distribution is the uniform distribution U_n over the domain, our new tester is surprisingly simple: to test whether p = U_n versus d_{TV} (p, U_n) >= epsilon, we simply threshold d_{TV}({p^}, U_n), where {p^} is the empirical probability distribution. The fact that this simple "plug-in" estimator is sample-optimal is surprising, even in the constant delta case. Indeed, it was believed that such a tester would not attain sublinear sample complexity even for constant values of epsilon and delta.
An important contribution of this work lies in the analysis techniques that we introduce in this context. First, we exploit an underlying strong convexity property to bound from below the expectation gap in the completeness and soundness cases. Second, we give a new, fast method for obtaining provably correct empirical estimates of the true worst-case failure probability for a broad class of uniformity testing statistics over all possible input distributions - including all previously studied statistics for this problem. We believe that our novel analysis techniques will be useful for other distribution testing problems as well.

Ilias Diakonikolas, Themis Gouleakis, John Peebles, and Eric Price. Sample-Optimal Identity Testing with High Probability. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{diakonikolas_et_al:LIPIcs.ICALP.2018.41, author = {Diakonikolas, Ilias and Gouleakis, Themis and Peebles, John and Price, Eric}, title = {{Sample-Optimal Identity Testing with High Probability}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.41}, URN = {urn:nbn:de:0030-drops-90459}, doi = {10.4230/LIPIcs.ICALP.2018.41}, annote = {Keywords: distribution testing, property testing, sample complexity} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

We study the question of testing structured properties (classes) of discrete distributions. Specifically, given sample access to an arbitrary distribution D over [n] and a property P, the goal is to distinguish between D in P and l_{1}(D,P)>epsilon. We develop a general algorithm for this question, which applies to a large range of "shape-constrained" properties, including monotone, log-concave, t-modal, piecewise-polynomial, and Poisson Binomial distributions. Moreover, for all cases considered, our algorithm has near-optimal sample complexity with regard to the domain size and is computationally efficient. For most of these classes, we provide the first non-trivial tester in the literature. In addition, we also describe a generic method to prove lower bounds for this problem, and use it to show our upper bounds are nearly tight. Finally, we extend some of our techniques to tolerant testing, deriving nearly-tight upper and lower bounds for the corresponding questions.

Clément L. Canonne, Ilias Diakonikolas, Themis Gouleakis, and Ronitt Rubinfeld. Testing Shape Restrictions of Discrete Distributions. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{canonne_et_al:LIPIcs.STACS.2016.25, author = {Canonne, Cl\'{e}ment L. and Diakonikolas, Ilias and Gouleakis, Themis and Rubinfeld, Ronitt}, title = {{Testing Shape Restrictions of Discrete Distributions}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {25:1--25:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.25}, URN = {urn:nbn:de:0030-drops-57260}, doi = {10.4230/LIPIcs.STACS.2016.25}, annote = {Keywords: property testing, probability distributions, statistics, lower bounds} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

The Lovasz Local Lemma (LLL) is a powerful tool that can be used to prove that an object having none of a set of bad properties exists, using the probabilistic method. In many applications of the LLL it is also desirable to explicitly construct the combinatorial object. Recently it was shown that this is possible using a randomized algorithm in the full asymmetric LLL setting [R. Moser and G. Tardos, 2010]. A strengthening of the LLL for the case of dense local neighborhoods proved in [R. Bissacot et al., 2010] was recently also made constructive in [W. Pegden, 2011]. In another recent work [B. Haupler, B. Saha, A. Srinivasan, 2010], it was proved that the algorithm of Moser and Tardos is still efficient even when the number of events is exponential. Here we prove that these last two contributions can be combined to yield a new version of the LLL.

Dimitris Achlioptas and Themis Gouleakis. Algorithmic Improvements of the Lovász Local Lemma via Cluster Expansion. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 16-23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{achlioptas_et_al:LIPIcs.FSTTCS.2012.16, author = {Achlioptas, Dimitris and Gouleakis, Themis}, title = {{Algorithmic Improvements of the Lov\'{a}sz Local Lemma via Cluster Expansion}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {16--23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.16}, URN = {urn:nbn:de:0030-drops-38440}, doi = {10.4230/LIPIcs.FSTTCS.2012.16}, annote = {Keywords: Probabilistic Method, Lov\'{a}sz Local Lemma, Algorithms} }

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