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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

This paper presents universal algorithms for clustering problems, including the widely studied k-median, k-means, and k-center objectives. The input is a metric space containing all potential client locations. The algorithm must select k cluster centers such that they are a good solution for any subset of clients that actually realize. Specifically, we aim for low regret, defined as the maximum over all subsets of the difference between the cost of the algorithm’s solution and that of an optimal solution. A universal algorithm’s solution sol for a clustering problem is said to be an (α, β)-approximation if for all subsets of clients C', it satisfies sol(C') ≤ α ⋅ opt(C') + β ⋅ mr, where opt(C') is the cost of the optimal solution for clients C' and mr is the minimum regret achievable by any solution.
Our main results are universal algorithms for the standard clustering objectives of k-median, k-means, and k-center that achieve (O(1), O(1))-approximations. These results are obtained via a novel framework for universal algorithms using linear programming (LP) relaxations. These results generalize to other 𝓁_p-objectives and the setting where some subset of the clients are fixed. We also give hardness results showing that (α, β)-approximation is NP-hard if α or β is at most a certain constant, even for the widely studied special case of Euclidean metric spaces. This shows that in some sense, (O(1), O(1))-approximation is the strongest type of guarantee obtainable for universal clustering.

Arun Ganesh, Bruce M. Maggs, and Debmalya Panigrahi. Universal Algorithms for Clustering Problems. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 70:1-70:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ganesh_et_al:LIPIcs.ICALP.2021.70, author = {Ganesh, Arun and Maggs, Bruce M. and Panigrahi, Debmalya}, title = {{Universal Algorithms for Clustering Problems}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {70:1--70:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.70}, URN = {urn:nbn:de:0030-drops-141397}, doi = {10.4230/LIPIcs.ICALP.2021.70}, annote = {Keywords: universal algorithms, clustering, k-median, k-means, k-center} }

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**Published in:** LIPIcs, Volume 192, 2nd Symposium on Foundations of Responsible Computing (FORC 2021)

We give the first closed-form privacy guarantees for the Generalized Gaussian mechanism (the mechanism that adds noise x to a vector with probability proportional to exp(-(||x||_p/σ)^p) for some σ, p), in the setting of answering k counting (i.e. sensitivity-1) queries about a database with (ε, δ)-differential privacy (in particular, with low 𝓁_∞-error). Just using Generalized Gaussian noise, we obtain a mechanism such that if the true answers to the queries are the vector d, the mechanism outputs answers d̃ with the 𝓁_∞-error guarantee:
𝔼[||d̃ - d||_∞] = O(√{k log log k log(1/δ)}/ε).
This matches the error bound of [Steinke and Ullman, 2017], but using a much simpler mechanism. By composing this mechanism with the sparse vector mechanism (generalizing a technique of [Steinke and Ullman, 2017]), we obtain a mechanism improving the √{k log log k} dependence on k to √{k log log log k}, Our main technical contribution is showing that certain powers of Generalized Gaussians, which follow a Generalized Gamma distribution, are sub-gamma.
In subsequent work, the optimal 𝓁_∞-error bound of O(√{k log (1/δ)}/ε) has been achieved by [Yuval Dagan and Gil Kur, 2020] and [Badih Ghazi et al., 2020] independently. However, the Generalized Gaussian mechanism has some qualitative advantages over the mechanisms used in these papers which may make it of interest to both practitioners and theoreticians, both in the setting of answering counting queries and more generally.

Arun Ganesh and Jiazheng Zhao. Privately Answering Counting Queries with Generalized Gaussian Mechanisms. In 2nd Symposium on Foundations of Responsible Computing (FORC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 192, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ganesh_et_al:LIPIcs.FORC.2021.1, author = {Ganesh, Arun and Zhao, Jiazheng}, title = {{Privately Answering Counting Queries with Generalized Gaussian Mechanisms}}, booktitle = {2nd Symposium on Foundations of Responsible Computing (FORC 2021)}, pages = {1:1--1:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-187-0}, ISSN = {1868-8969}, year = {2021}, volume = {192}, editor = {Ligett, Katrina and Gupta, Swati}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2021.1}, URN = {urn:nbn:de:0030-drops-138698}, doi = {10.4230/LIPIcs.FORC.2021.1}, annote = {Keywords: Differential privacy, counting queries, Generalized Gaussians} }

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**Published in:** LIPIcs, Volume 172, 20th International Workshop on Algorithms in Bioinformatics (WABI 2020)

We consider the following model for sampling pairs of strings: s₁ is a uniformly random bitstring of length n, and s₂ is the bitstring arrived at by applying substitutions, insertions, and deletions to each bit of s₁ with some probability. We show that the edit distance between s₁ and s₂ can be computed in O(n ln n) time with high probability, as long as each bit of s₁ has a mutation applied to it with probability at most a small constant. The algorithm is simple and only uses the textbook dynamic programming algorithm as a primitive, first computing an approximate alignment between the two strings, and then running the dynamic programming algorithm restricted to entries close to the approximate alignment. The analysis of our algorithm provides theoretical justification for alignment heuristics used in practice such as BLAST, FASTA, and MAFFT, which also start by computing approximate alignments quickly and then find the best alignment near the approximate alignment. Our main technical contribution is a partitioning of alignments such that the number of the subsets in the partition is not too large and every alignment in one subset is worse than an alignment considered by our algorithm with high probability. Similar techniques may be of interest in the average-case analysis of other problems commonly solved via dynamic programming.

Arun Ganesh and Aaron Sy. Near-Linear Time Edit Distance for Indel Channels. In 20th International Workshop on Algorithms in Bioinformatics (WABI 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 172, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ganesh_et_al:LIPIcs.WABI.2020.17, author = {Ganesh, Arun and Sy, Aaron}, title = {{Near-Linear Time Edit Distance for Indel Channels}}, booktitle = {20th International Workshop on Algorithms in Bioinformatics (WABI 2020)}, pages = {17:1--17:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-161-0}, ISSN = {1868-8969}, year = {2020}, volume = {172}, editor = {Kingsford, Carl and Pisanti, Nadia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2020.17}, URN = {urn:nbn:de:0030-drops-128061}, doi = {10.4230/LIPIcs.WABI.2020.17}, annote = {Keywords: edit distance, average-case analysis, dynamic programming, sequence alignment} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Robust optimization is a widely studied area in operations research, where the algorithm takes as input a range of values and outputs a single solution that performs well for the entire range. Specifically, a robust algorithm aims to minimize regret, defined as the maximum difference between the solution’s cost and that of an optimal solution in hindsight once the input has been realized. For graph problems in P, such as shortest path and minimum spanning tree, robust polynomial-time algorithms that obtain a constant approximation on regret are known. In this paper, we study robust algorithms for minimizing regret in NP-hard graph optimization problems, and give constant approximations on regret for the classical traveling salesman and Steiner tree problems.

Arun Ganesh, Bruce M. Maggs, and Debmalya Panigrahi. Robust Algorithms for TSP and Steiner Tree. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 54:1-54:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ganesh_et_al:LIPIcs.ICALP.2020.54, author = {Ganesh, Arun and Maggs, Bruce M. and Panigrahi, Debmalya}, title = {{Robust Algorithms for TSP and Steiner Tree}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {54:1--54:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.54}, URN = {urn:nbn:de:0030-drops-124619}, doi = {10.4230/LIPIcs.ICALP.2020.54}, annote = {Keywords: Robust optimization, Steiner tree, traveling salesman problem} }

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