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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We identify several genres of search problems beyond NP for which existence of solutions is guaranteed. One class that seems especially rich in such problems is PEPP (for "polynomial empty pigeonhole principle"), which includes problems related to existence theorems proved through the union bound, such as finding a bit string that is far from all codewords, finding an explicit rigid matrix, as well as a problem we call Complexity, capturing Complexity Theory’s quest. When the union bound is generous, in that solutions constitute at least a polynomial fraction of the domain, we have a family of seemingly weaker classes α-PEPP, which are inside FP^NP|poly. Higher in the hierarchy, we identify the constructive version of the Sauer-Shelah lemma and the appropriate generalization of PPP that contains it, as well as the problem of finding a king in a tournament (a vertex k such that all other vertices are defeated by k, or by somebody k defeated).

Robert Kleinberg, Oliver Korten, Daniel Mitropolsky, and Christos Papadimitriou. Total Functions in the Polynomial Hierarchy. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 44:1-44:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kleinberg_et_al:LIPIcs.ITCS.2021.44, author = {Kleinberg, Robert and Korten, Oliver and Mitropolsky, Daniel and Papadimitriou, Christos}, title = {{Total Functions in the Polynomial Hierarchy}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {44:1--44:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.44}, URN = {urn:nbn:de:0030-drops-135835}, doi = {10.4230/LIPIcs.ITCS.2021.44}, annote = {Keywords: total complexity, polynomial hierarchy, pigeonhole principle} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

The following is a summary of the paper "Inferential Privacy Guarantees for Differentially Private Mechanisms", presented at the Eighth Innovations in Theoretical Computer Science Conference in January 2017. The full version of the paper can be found on arXiv at the URL https://arxiv.org/abs/1603.01508.

Arpita Ghosh and Robert Kleinberg. Inferential Privacy Guarantees for Differentially Private Mechanisms. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 9:1-9:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ghosh_et_al:LIPIcs.ITCS.2017.9, author = {Ghosh, Arpita and Kleinberg, Robert}, title = {{Inferential Privacy Guarantees for Differentially Private Mechanisms}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {9:1--9:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.9}, URN = {urn:nbn:de:0030-drops-81451}, doi = {10.4230/LIPIcs.ITCS.2017.9}, annote = {Keywords: differential privacy, statistical inference, statistical mechanics} }

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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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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Motivated by applications in computer vision and databases, we introduce and study the Simultaneous Nearest Neighbor Search (SNN) problem. Given a set of data points, the goal of SNN is to design a data structure that, given a collection of queries, finds a collection of close points that are compatible with each other. Formally, we are given k query points Q=q_1,...,q_k, and a compatibility graph G with vertices in Q, and the goal is to return data points p_1,...,p_k that minimize (i) the weighted sum of the distances from q_i to p_i and (ii) the weighted sum, over all edges (i,j) in the compatibility graph G, of the distances between p_i and p_j. The problem has several applications in computer vision and databases, where one wants to return a set of *consistent* answers to multiple related queries. Furthermore, it generalizes several well-studied computational problems, including Nearest Neighbor Search, Aggregate Nearest Neighbor Search and the 0-extension problem.
In this paper we propose and analyze the following general two-step method for designing efficient data structures for SNN. In the first step, for each query point q_i we find its (approximate) nearest neighbor point p'_i; this can be done efficiently using existing approximate nearest neighbor structures. In the second step, we solve an off-line optimization problem over sets q_1,...,q_k and p'_1,...,p'_k; this can be done efficiently given that k is much smaller than n. Even though p'_1,...,p'_k might not constitute the optimal answers to queries q_1,...,q_k, we show that, for the unweighted case, the resulting algorithm satisfies a O(log k/log log k)-approximation guarantee. Furthermore, we show that the approximation factor can be in fact reduced to a constant for compatibility graphs frequently occurring in practice, e.g., 2D grids, 3D grids or planar graphs.
Finally, we validate our theoretical results by preliminary experiments. In particular, we show that the empirical approximation factor provided by the above approach is very close to 1.

Piotr Indyk, Robert Kleinberg, Sepideh Mahabadi, and Yang Yuan. Simultaneous Nearest Neighbor Search. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 44:1-44:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{indyk_et_al:LIPIcs.SoCG.2016.44, author = {Indyk, Piotr and Kleinberg, Robert and Mahabadi, Sepideh and Yuan, Yang}, title = {{Simultaneous Nearest Neighbor Search}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.44}, URN = {urn:nbn:de:0030-drops-59360}, doi = {10.4230/LIPIcs.SoCG.2016.44}, annote = {Keywords: Approximate Nearest Neighbor, Metric Labeling, 0-extension, Simultaneous Nearest Neighbor, Group Nearest Neighbor} }

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

Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions f1, f2 and f3, mapping {0,1}^k to {0,1}, are said to be triangle free if there is no x, y in {0,1}^k such that f1(x) = f2(y) = f3(x + y) = 1. This property is known to be strongly testable (Green 2005), but the number of queries needed is upper-bounded only by a tower of twos whose height is polynomial in 1 / epsislon, where epsislon is the distance between the tested function triple and triangle-freeness, i.e., the minimum fraction of function values that need to be modified to make the triple triangle free. A lower bound of (1 / epsilon)^2.423 for any one-sided tester was given by Bhattacharyya and Xie (2010). In this work we improve this bound to (1 / epsilon)^6.619. Interestingly, we prove this by way of a combinatorial construction called uniquely solvable puzzles that was at the heart of Coppersmith and Winograd's renowned matrix multiplication algorithm.

Hu Fu and Robert Kleinberg. Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 669-676, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{fu_et_al:LIPIcs.APPROX-RANDOM.2014.669, author = {Fu, Hu and Kleinberg, Robert}, title = {{Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {669--676}, 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.2014.669}, URN = {urn:nbn:de:0030-drops-47304}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.669}, annote = {Keywords: Property testing, linear invariance, fast matrix multiplication, uniquely solvable puzzles} }

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