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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Differentially private algorithms protect individuals in data analysis scenarios by ensuring that there is only a weak correlation between the existence of the user in the data and the result of the analysis. Dynamic graph algorithms maintain the solution to a problem (e.g., a matching) on an evolving input, i.e., a graph where nodes or edges are inserted or deleted over time. They output the value of the solution after each update operation, i.e., continuously. We study (event-level and user-level) differentially private algorithms for graph problems under continual observation, i.e., differentially private dynamic graph algorithms. We present event-level private algorithms for partially dynamic counting-based problems such as triangle count that improve the additive error by a polynomial factor (in the length T of the update sequence) on the state of the art, resulting in the first algorithms with additive error polylogarithmic in T.
We also give ε-differentially private and partially dynamic algorithms for minimum spanning tree, minimum cut, densest subgraph, and maximum matching. The additive error of our improved MST algorithm is O(W log^{3/2}T / ε), where W is the maximum weight of any edge, which, as we show, is tight up to a (√{log T} / ε)-factor. For the other problems, we present a partially-dynamic algorithm with multiplicative error (1+β) for any constant β > 0 and additive error O(W log(nW) log(T) / (ε β)). Finally, we show that the additive error for a broad class of dynamic graph algorithms with user-level privacy must be linear in the value of the output solution’s range.

Hendrik Fichtenberger, Monika Henzinger, and Lara Ost. Differentially Private Algorithms for Graphs Under Continual Observation. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 42:1-42:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fichtenberger_et_al:LIPIcs.ESA.2021.42, author = {Fichtenberger, Hendrik and Henzinger, Monika and Ost, Lara}, title = {{Differentially Private Algorithms for Graphs Under Continual Observation}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {42:1--42:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.42}, URN = {urn:nbn:de:0030-drops-146230}, doi = {10.4230/LIPIcs.ESA.2021.42}, annote = {Keywords: differential privacy, continual observation, dynamic graph algorithms} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We consider the fundamental question of understanding the relative power of two important computational models: property testing and data streaming. We present a novel framework closely linking these areas in the setting of general graphs in the context of constant-query complexity testing and constant-space streaming. Our main result is a generic transformation of a one-sided error property tester in the random-neighbor model with constant query complexity into a one-sided error property tester in the streaming model with constant space complexity. Previously such a generic transformation was only known for bounded-degree graphs.

Artur Czumaj, Hendrik Fichtenberger, Pan Peng, and Christian Sohler. Testable Properties in General Graphs and Random Order Streaming. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{czumaj_et_al:LIPIcs.APPROX/RANDOM.2020.16, author = {Czumaj, Artur and Fichtenberger, Hendrik and Peng, Pan and Sohler, Christian}, title = {{Testable Properties in General Graphs and Random Order Streaming}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {16:1--16:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.16}, URN = {urn:nbn:de:0030-drops-126190}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.16}, annote = {Keywords: Graph property testing, sublinear algorithms, graph streaming algorithms} }

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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)

We present a simple sublinear-time algorithm for sampling an arbitrary subgraph H exactly uniformly from a graph G, to which the algorithm has access by performing the following types of queries: (1) uniform vertex queries, (2) degree queries, (3) neighbor queries, (4) pair queries and (5) edge sampling queries. The query complexity and running time of our algorithm are Õ(min{m, (m^ρ(H))/#H}) and Õ((m^ρ(H))/#H}), respectively, where ρ(H) is the fractional edge-cover of H and #H is the number of copies of H in G. For any clique on r vertices, i.e., H = K_r, our algorithm is almost optimal as any algorithm that samples an H from any distribution that has Ω(1) total probability mass on the set of all copies of H must perform Ω(min{m, (m^ρ(H))/(#H⋅(cr)^r)}) queries.
Together with the query and time complexities of the (1±ε)-approximation algorithm for the number of subgraphs H by Assadi et al. [Sepehr Assadi et al., 2018] and the lower bound by Eden and Rosenbaum [Eden and Rosenbaum, 2018] for approximately counting cliques, our results suggest that in our query model, approximately counting cliques is "equivalent to" exactly uniformly sampling cliques, in the sense that the query and time complexities of exactly uniform sampling and randomized approximate counting are within polylogarithmic factor of each other. This stands in interesting contrast to an analogous relation between approximate counting and almost uniformly sampling for self-reducible problems in the polynomial-time regime by Jerrum, Valiant and Vazirani [Jerrum et al., 1986].

Hendrik Fichtenberger, Mingze Gao, and Pan Peng. Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fichtenberger_et_al:LIPIcs.ICALP.2020.45, author = {Fichtenberger, Hendrik and Gao, Mingze and Peng, Pan}, title = {{Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {45:1--45:13}, 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.2020.45}, URN = {urn:nbn:de:0030-drops-124526}, doi = {10.4230/LIPIcs.ICALP.2020.45}, annote = {Keywords: Graph sampling, Graph algorithms, Sublinear-time algorithms} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We study property testing in the distributed model and extend its setting from testing with one-sided error to testing with two-sided error. In particular, we develop a two-sided error property tester for general graphs with round complexity O(log(n) / (epsilon Phi^2)) in the CONGEST model, which accepts graphs with conductance Phi and rejects graphs that are epsilon-far from having conductance at least Phi^2 / 1000 with constant probability. Our main insight is that one can start poly(n) random walks from a few random vertices without violating the congestion and unite the results to obtain a consistent answer from all vertices. For connected graphs, this is even possible when the number of vertices is unknown. We also obtain a matching Omega(log n) lower bound for the LOCAL and CONGEST models by an indistinguishability argument. Although the power of vertex labels that arises from two-sided error might seem to be much stronger than in the sequential query model, we can show that this is not the case.

Hendrik Fichtenberger and Yadu Vasudev. A Two-Sided Error Distributed Property Tester For Conductance. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fichtenberger_et_al:LIPIcs.MFCS.2018.19, author = {Fichtenberger, Hendrik and Vasudev, Yadu}, title = {{A Two-Sided Error Distributed Property Tester For Conductance}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul 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.MFCS.2018.19}, URN = {urn:nbn:de:0030-drops-96019}, doi = {10.4230/LIPIcs.MFCS.2018.19}, annote = {Keywords: property testing, distributed algorithms, conductance} }

Document

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

We consider one-sided error property testing of F-minor freeness in bounded-degree graphs for any finite family of graphs F that contains a minor of K_{2,k}, the k-circus graph, or the (k x 2)-grid for any k in N. This includes, for instance, testing whether a graph is outerplanar or a cactus graph. The query complexity of our algorithm in terms of the number of vertices in the graph, n, is O~(n^{2/3} / epsilon^5). Czumaj et al. (2014) showed that cycle-freeness and C_k-minor freeness can be tested with query complexity O~(sqrt{n}) by using random walks, and that testing H-minor freeness for any H that contains a cycles requires Omega(sqrt{n}) queries. In contrast to these results, we analyze the structure of the graph and show that either we can find a subgraph of sublinear size that includes the forbidden minor H, or we can find a pair of disjoint subsets of vertices whose edge-cut is large, which induces an H-minor.

Hendrik Fichtenberger, Reut Levi, Yadu Vasudev, and Maximilian Wötzel. A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fichtenberger_et_al:LIPIcs.ICALP.2018.52, author = {Fichtenberger, Hendrik and Levi, Reut and Vasudev, Yadu and W\"{o}tzel, Maximilian}, title = {{A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {52:1--52:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.52}, URN = {urn:nbn:de:0030-drops-90563}, doi = {10.4230/LIPIcs.ICALP.2018.52}, annote = {Keywords: graph property testing, minor-free graphs} }

Document

**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Let G=(V,E) be an undirected graph with maximum degree d. The k-disc of a vertex v is defined as the rooted subgraph that is induced by all vertices whose distance to v is at most k. The k-disc frequency vector of G, freq(G), is a vector indexed by all isomorphism types of k-discs. For each such isomorphism type Gamma, the k-disc frequency vector counts the fraction of vertices that have k-disc isomorphic to Gamma. Thus, the frequency vector freq(G) of G captures the local structure of G. A natural question is whether one can construct a much smaller graph H such that H has a similar local structure. N. Alon proved that for any epsilon>0 there always exists a graph H whose size is independent of |V| and whose frequency vector satisfies ||freq(G) - freq(G)||_1 <= epsilon. However, his proof is only existential and neither gives an explicit bound on the size of H nor an efficient algorithm. He gave the open problem to find such explicit bounds. In this paper, we solve this problem for the special case of high girth graphs. We show how to efficiently compute a graph H with the above properties when G has girth at least 2k+2 and we give explicit bounds on the size of H.

Hendrik Fichtenberger, Pan Peng, and Christian Sohler. On Constant-Size Graphs That Preserve the Local Structure of High-Girth Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 786-799, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{fichtenberger_et_al:LIPIcs.APPROX-RANDOM.2015.786, author = {Fichtenberger, Hendrik and Peng, Pan and Sohler, Christian}, title = {{On Constant-Size Graphs That Preserve the Local Structure of High-Girth Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {786--799}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.786}, URN = {urn:nbn:de:0030-drops-53363}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.786}, annote = {Keywords: local graph structure, k-disc frequency vector, graph property testing} }

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