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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We study the problem of estimating the number of edges in an n-vertex graph, accessed via the Bipartite Independent Set query model introduced by Beame et al. (TALG '20). In this model, each query returns a Boolean, indicating the existence of at least one edge between two specified sets of nodes. We present a non-adaptive algorithm that returns a (1± ε) relative error approximation to the number of edges, with query complexity Õ(ε^{-5}log⁵ n), where Õ(⋅) hides poly(log log n) dependencies. This is the first non-adaptive algorithm in this setting achieving poly(1/ε,log n) query complexity. Prior work requires Ω(log² n) rounds of adaptivity. We avoid this by taking a fundamentally different approach, inspired by work on single-pass streaming algorithms. Moreover, for constant ε, our query complexity significantly improves on the best known adaptive algorithm due to Bhattacharya et al. (STACS '22), which requires O(ε^{-2} log^{11} n) queries. Building on our edge estimation result, we give the first {non-adaptive} algorithm for outputting a nearly uniformly sampled edge with query complexity Õ(ε^{-6} log⁶ n), improving on the works of Dell et al. (SODA '20) and Bhattacharya et al. (STACS '22), which require Ω(log³ n) rounds of adaptivity. Finally, as a consequence of our edge sampling algorithm, we obtain a Õ(n log^8 n) query algorithm for connectivity, using two rounds of adaptivity. This improves on a three-round algorithm of Assadi et al. (ESA '21) and is tight; there is no non-adaptive algorithm for connectivity making o(n²) queries.

Raghavendra Addanki, Andrew McGregor, and Cameron Musco. Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{addanki_et_al:LIPIcs.ESA.2022.2, author = {Addanki, Raghavendra and McGregor, Andrew and Musco, Cameron}, title = {{Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {2:1--2:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.2}, URN = {urn:nbn:de:0030-drops-169400}, doi = {10.4230/LIPIcs.ESA.2022.2}, annote = {Keywords: sublinear graph algorithms, bipartite independent set queries, edge sampling and counting, graph connectivity, query adaptivity} }

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**Published in:** LIPIcs, Volume 220, 25th International Conference on Database Theory (ICDT 2022)

Given an n-point metric space ({𝒳},d) where each point belongs to one of m = O(1) different categories or groups and a set of integers k₁, …, k_m, the fair Max-Min diversification problem is to select k_i points belonging to category i ∈ [m], such that the minimum pairwise distance between selected points is maximized. The problem was introduced by Moumoulidou et al. [ICDT 2021] and is motivated by the need to down-sample large data sets in various applications so that the derived sample achieves a balance over diversity, i.e., the minimum distance between a pair of selected points, and fairness, i.e., ensuring enough points of each category are included. We prove the following results:
1) We first consider general metric spaces. We present a randomized polynomial time algorithm that returns a factor 2-approximation to the diversity but only satisfies the fairness constraints in expectation. Building upon this result, we present a 6-approximation that is guaranteed to satisfy the fairness constraints up to a factor 1-ε for any constant ε. We also present a linear time algorithm returning an m+1 approximation with exact fairness. The best previous result was a 3m-1 approximation.
2) We then focus on Euclidean metrics. We first show that the problem can be solved exactly in one dimension. {For constant dimensions, categories and any constant ε > 0, we present a 1+ε approximation algorithm that runs in O(nk) + 2^{O(k)} time where k = k₁+…+k_m.} We can improve the running time to O(nk)+poly(k) at the expense of only picking (1-ε) k_i points from category i ∈ [m].
Finally, we present algorithms suitable to processing massive data sets including single-pass data stream algorithms and composable coresets for the distributed processing.

Raghavendra Addanki, Andrew McGregor, Alexandra Meliou, and Zafeiria Moumoulidou. Improved Approximation and Scalability for Fair Max-Min Diversification. In 25th International Conference on Database Theory (ICDT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 220, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{addanki_et_al:LIPIcs.ICDT.2022.7, author = {Addanki, Raghavendra and McGregor, Andrew and Meliou, Alexandra and Moumoulidou, Zafeiria}, title = {{Improved Approximation and Scalability for Fair Max-Min Diversification}}, booktitle = {25th International Conference on Database Theory (ICDT 2022)}, pages = {7:1--7:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-223-5}, ISSN = {1868-8969}, year = {2022}, volume = {220}, editor = {Olteanu, Dan and Vortmeier, Nils}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2022.7}, URN = {urn:nbn:de:0030-drops-158812}, doi = {10.4230/LIPIcs.ICDT.2022.7}, annote = {Keywords: algorithmic fairness, diversity maximization, data selection, approximation algorithms} }

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