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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of P to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity.

Pavel Hrubeš and Amir Yehudayoff. Shadows of Newton Polytopes. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hrubes_et_al:LIPIcs.CCC.2021.9, author = {Hrube\v{s}, Pavel and Yehudayoff, Amir}, title = {{Shadows of Newton Polytopes}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {9:1--9:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.9}, URN = {urn:nbn:de:0030-drops-142833}, doi = {10.4230/LIPIcs.CCC.2021.9}, annote = {Keywords: Newton polytope, Monotone arithmetic circuit} }

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

We consider the following question: using a source of labeled data and interaction with an untrusted prover, what is the complexity of verifying that a given hypothesis is "approximately correct"? We study interactive proof systems for PAC verification, where a verifier that interacts with a prover is required to accept good hypotheses, and reject bad hypotheses. Both the verifier and the prover are efficient and have access to labeled data samples from an unknown distribution. We are interested in cases where the verifier can use significantly less data than is required for (agnostic) PAC learning, or use a substantially cheaper data source (e.g., using only random samples for verification, even though learning requires membership queries). We believe that today, when data and data-driven algorithms are quickly gaining prominence, the question of verifying purported outcomes of data analyses is very well-motivated.
We show three main results. First, we prove that for a specific hypothesis class, verification is significantly cheaper than learning in terms of sample complexity, even if the verifier engages with the prover only in a single-round (NP-like) protocol. Moreover, for this class we prove that single-round verification is also significantly cheaper than testing closeness to the class. Second, for the broad class of Fourier-sparse boolean functions, we show a multi-round (IP-like) verification protocol, where the prover uses membership queries, and the verifier is able to assess the result while only using random samples. Third, we show that verification is not always more efficient. Namely, we show a class of functions where verification requires as many samples as learning does, up to a logarithmic factor.

Shafi Goldwasser, Guy N. Rothblum, Jonathan Shafer, and Amir Yehudayoff. Interactive Proofs for Verifying Machine Learning. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goldwasser_et_al:LIPIcs.ITCS.2021.41, author = {Goldwasser, Shafi and Rothblum, Guy N. and Shafer, Jonathan and Yehudayoff, Amir}, title = {{Interactive Proofs for Verifying Machine Learning}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {41:1--41:19}, 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.2021.41}, URN = {urn:nbn:de:0030-drops-135806}, doi = {10.4230/LIPIcs.ITCS.2021.41}, annote = {Keywords: PAC learning, Fourier analysis of boolean functions, Complexity gaps, Complexity lower bounds, Goldreich-Levin algorithm, Kushilevitz-Mansour algorithm, Distribution testing} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets S_1,...,S_k subset [n] is balancing if for every subset X subset {1,2,...,n} of size n/2, there is an i in [k] so that |S_i cap X| = |S_i|/2. We extend and simplify the framework developed by Hegedűs for proving lower bounds on the size of balancing set families. We prove that if n=2p for a prime p, then k >= p. For arbitrary values of n, we show that k >= n/2 - o(n).
We then exploit the connection between balancing families and depth-2 threshold circuits. This connection helps resolve a question raised by Kulikov and Podolskii on the fan-in of depth-2 majority circuits computing the majority function on n bits. We show that any depth-2 threshold circuit that computes the majority on n bits has at least one gate with fan-in at least n/2 - o(n). We also prove a sharp lower bound on the fan-in of depth-2 threshold circuits computing a specific weighted threshold function.

Pavel Hrubeš, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, and Amir Yehudayoff. Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 72:1-72:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hrubes_et_al:LIPIcs.ICALP.2019.72, author = {Hrube\v{s}, Pavel and Natarajan Ramamoorthy, Sivaramakrishnan and Rao, Anup and Yehudayoff, Amir}, title = {{Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {72:1--72:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.72}, URN = {urn:nbn:de:0030-drops-106487}, doi = {10.4230/LIPIcs.ICALP.2019.72}, annote = {Keywords: Balancing sets, depth-2 threshold circuits, polynomials, majority, weighted thresholds} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove an amplification result for weak epsilon-nets.

Shay Moran and Amir Yehudayoff. On Weak epsilon-Nets and the Radon Number. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{moran_et_al:LIPIcs.SoCG.2019.51, author = {Moran, Shay and Yehudayoff, Amir}, title = {{On Weak epsilon-Nets and the Radon Number}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {51:1--51:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.51}, URN = {urn:nbn:de:0030-drops-104551}, doi = {10.4230/LIPIcs.SoCG.2019.51}, annote = {Keywords: abstract convexity, weak epsilon nets, Radon number, VC dimension, Haussler packing lemma, Kneser graphs} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Key-agreement protocols whose security is proven in the random oracle model are an important alternative to protocols based on public-key cryptography. In the random oracle model, the parties and the eavesdropper have access to a shared random function (an "oracle"), but the parties are limited in the number of queries they can make to the oracle. The random oracle serves as an abstraction for black-box access to a symmetric cryptographic primitive, such as a collision resistant hash. Unfortunately, as shown by Impagliazzo and Rudich [STOC '89] and Barak and Mahmoody [Crypto '09], such protocols can only guarantee limited secrecy: the key of any l-query protocol can be revealed by an O(l^2)-query adversary. This quadratic gap between the query complexity of the honest parties and the eavesdropper matches the gap obtained by the Merkle's Puzzles protocol of Merkle [CACM '78].
In this work we tackle a new aspect of key-agreement protocols in the random oracle model: their communication complexity. In Merkle's Puzzles, to obtain secrecy against an eavesdropper that makes roughly l^2 queries, the honest parties need to exchange Omega(l) bits. We show that for protocols with certain natural properties, ones that Merkle's Puzzle has, such high communication is unavoidable. Specifically, this is the case if the honest parties' queries are uniformly random, or alternatively if the protocol uses non-adaptive queries and has only two rounds. Our proof for the first setting uses a novel reduction from the set-disjointness problem in two-party communication complexity. For the second setting we prove the lower bound directly, using information-theoretic arguments.
Understanding the communication complexity of protocols whose security is proven (in the random-oracle model) is an important question in the study of practical protocols. Our results and proof techniques are a first step in this direction.

Iftach Haitner, Noam Mazor, Rotem Oshman, Omer Reingold, and Amir Yehudayoff. On the Communication Complexity of Key-Agreement Protocols. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{haitner_et_al:LIPIcs.ITCS.2019.40, author = {Haitner, Iftach and Mazor, Noam and Oshman, Rotem and Reingold, Omer and Yehudayoff, Amir}, title = {{On the Communication Complexity of Key-Agreement Protocols}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {40:1--40:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.40}, URN = {urn:nbn:de:0030-drops-101335}, doi = {10.4230/LIPIcs.ITCS.2019.40}, annote = {Keywords: key agreement, random oracle, communication complexity, Merkle's puzzles} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

The isoperimetric profile of a graph is a function that measures, for an integer k, the size of the smallest edge boundary over all sets of vertices of size k. We observe a connection between isoperimetric profiles and computational complexity. We illustrate this connection by an example from communication complexity, but our main result is in algebraic complexity.
We prove a sharp super-polynomial separation between monotone arithmetic circuits and monotone arithmetic branching programs. This shows that the classical simulation of arithmetic circuits by arithmetic branching programs by Valiant, Skyum, Berkowitz, and Rackoff (1983) cannot be improved, as long as it preserves monotonicity.
A key ingredient in the proof is an accurate analysis of the isoperimetric profile of finite full binary trees. We show that the isoperimetric profile of a full binary tree constantly fluctuates between one and almost the depth of the tree.

Pavel Hrubes and Amir Yehudayoff. On Isoperimetric Profiles and Computational Complexity. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 89:1-89:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hrubes_et_al:LIPIcs.ICALP.2016.89, author = {Hrubes, Pavel and Yehudayoff, Amir}, title = {{On Isoperimetric Profiles and Computational Complexity}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {89:1--89:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.89}, URN = {urn:nbn:de:0030-drops-61964}, doi = {10.4230/LIPIcs.ICALP.2016.89}, annote = {Keywords: Monotone computation, separations, communication complexity, isoperimetry} }

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

We study internal compression of communication protocols to their internal entropy, which is the entropy of the transcript from the players' perspective. We provide two internal compression schemes with error. One of a protocol of Feige et al. for finding the first difference between two strings. The second and main one is an internal compression with error epsilon > 0 of a protocol with internal entropy H^{int} and communication complexity C to a protocol with communication at most order (H^{int}/epsilon)^2 * log(log(C)).
This immediately implies a similar compression to the internal information of public-coin protocols, which provides an exponential improvement over previously known public-coin compressions in the dependence on C. It further shows that in a recent protocol of Ganor, Kol and Raz, it is impossible to move the private randomness to be public without an exponential cost. To the best of our knowledge, No such example was previously known.

Balthazar Bauer, Shay Moran, and Amir Yehudayoff. Internal Compression of Protocols to Entropy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 481-496, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bauer_et_al:LIPIcs.APPROX-RANDOM.2015.481, author = {Bauer, Balthazar and Moran, Shay and Yehudayoff, Amir}, title = {{Internal Compression of Protocols to Entropy}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {481--496}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.481}, URN = {urn:nbn:de:0030-drops-53198}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.481}, annote = {Keywords: Communication complexity, Information complexity, Compression, Simulation, Entropy} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

We show that the deterministic number-on-forehead communication complexity of set disjointness for k parties on a universe of size n is Omega(n/4^k). This gives the first lower bound that is linear in n, nearly matching Grolmusz's upper bound of O(log^2(n) + k^2n/2^k). We also simplify the proof of Sherstov's Omega(sqrt(n)/(k2^k)) lower bound for the randomized communication complexity of set disjointness.

Anup Rao and Amir Yehudayoff. Simplified Lower Bounds on the Multiparty Communication Complexity of Disjointness. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 88-101, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{rao_et_al:LIPIcs.CCC.2015.88, author = {Rao, Anup and Yehudayoff, Amir}, title = {{Simplified Lower Bounds on the Multiparty Communication Complexity of Disjointness}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {88--101}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.88}, URN = {urn:nbn:de:0030-drops-50769}, doi = {10.4230/LIPIcs.CCC.2015.88}, annote = {Keywords: communication complexity, set disjointness, number on forehead, lower bounds} }

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