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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Embedding graphs in a geographical or latent space, i.e. inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r = n^α for α > 0, with high probability reconstructs the vertex positions with a maximum error of O(n^β) where β = 1/2-(4/3)α, until α ≥ 3/8 where β = 0 and the error becomes O(√{log n}). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in ℝ³ and to hypercubes in any constant dimension.

Varsha Dani, Josep Díaz, Thomas P. Hayes, and Cristopher Moore. Improved Reconstruction of Random Geometric Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dani_et_al:LIPIcs.ICALP.2022.48, author = {Dani, Varsha and D{\'\i}az, Josep and Hayes, Thomas P. and Moore, Cristopher}, title = {{Improved Reconstruction of Random Geometric Graphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {48:1--48:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.48}, URN = {urn:nbn:de:0030-drops-163897}, doi = {10.4230/LIPIcs.ICALP.2022.48}, annote = {Keywords: Reconstruction algorithm, distances in RGG, d-dimensional hypercube, 3 dimensional sphere} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions from supposedly-intractable problems to MCSP / MKTP hinged on the power of MCSP / MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of Graph Isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include Linear Code Equivalence, Permutation Group Conjugacy, and Matrix Subspace Conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest.

Eric Allender, Joshua A. Grochow, Dieter van Melkebeek, Cristopher Moore, and Andrew Morgan. Minimum Circuit Size, Graph Isomorphism, and Related Problems. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{allender_et_al:LIPIcs.ITCS.2018.20, author = {Allender, Eric and Grochow, Joshua A. and van Melkebeek, Dieter and Moore, Cristopher and Morgan, Andrew}, title = {{Minimum Circuit Size, Graph Isomorphism, and Related Problems}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {20:1--20:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.20}, URN = {urn:nbn:de:0030-drops-83455}, doi = {10.4230/LIPIcs.ITCS.2018.20}, annote = {Keywords: Reductions between NP-intermediate problems, Graph Isomorphism, Minimum Circuit Size Problem, time-bounded Kolmogorov complexity} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

We derive upper and lower bounds on the degree d for which the Lovasz theta function, or equivalently sum-of-squares proofs with degree two, can refute the existence of a k-coloring in random regular graphs G(n,d). We show that this type of refutation fails well above the k-colorability transition, and in particular everywhere below the Kesten-Stigum threshold. This is consistent with the conjecture that refuting k-colorability, or distinguishing G(n,d) from the planted coloring model, is hard in this region. Our results also apply to the disassortative case of the stochastic block model, adding evidence to the conjecture that there is a regime where community detection is computationally hard even though it is information-theoretically possible. Using orthogonal polynomials, we also provide explicit upper bounds on the theta function for regular graphs of a given girth, which may be of independent interest.

Jess Banks, Robert Kleinberg, and Cristopher Moore. The Lovász Theta Function for Random Regular Graphs and Community Detection in the Hard Regime. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 28:1-28:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{banks_et_al:LIPIcs.APPROX-RANDOM.2017.28, author = {Banks, Jess and Kleinberg, Robert and Moore, Cristopher}, title = {{The Lov\'{a}sz Theta Function for Random Regular Graphs and Community Detection in the Hard Regime}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {28:1--28:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.28}, URN = {urn:nbn:de:0030-drops-75771}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.28}, annote = {Keywords: Lov\'{a}sz Theta Function, Random Regular Graphs, Sum of Squares, Orthogonal Polynomials} }

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Complete Volume

**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

LIPIcs, Volume 28, APPROX/RANDOM'14, Complete Volume

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@Proceedings{jansen_et_al:LIPIcs.APPROX-RANDOM.2014, title = {{LIPIcs, Volume 28, APPROX/RANDOM'14, Complete Volume}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014}, URN = {urn:nbn:de:0030-drops-47603}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014}, annote = {Keywords: Network Architecture and Design, Computer-communication, Coding and Information Theory, Theory of Computation, Computation by Abstract Devices, Models of Computation – relations between models, Modes of Computation, Complexity Measures and Classes, Analysis of Algorithms and Problem Complexity} }

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Front Matter

**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Frontmatter, Table of Contents, Preface, Conference Organization

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. i-xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{jansen_et_al:LIPIcs.APPROX-RANDOM.2014.i, author = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, title = {{Frontmatter, Table of Contents, Preface, Conference Organization}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {i--xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.i}, URN = {urn:nbn:de:0030-drops-46846}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.i}, annote = {Keywords: Frontmatter, Table of Contents, Preface, Conference Organization} }