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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We study the problem of performing counting queries at different levels in hierarchical structures while preserving individuals' privacy. Motivated by applications, we propose a new error measure for this problem by considering a combination of multiplicative and additive approximation to the query results. We examine known mechanisms in differential privacy (DP) and prove their optimality, under this measure, in the pure-DP setting. In the approximate-DP setting, we design new algorithms achieving significant improvements over known ones.

Badih Ghazi, Pritish Kamath, Ravi Kumar, Pasin Manurangsi, and Kewen Wu. On Differentially Private Counting on Trees. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 66:1-66:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ghazi_et_al:LIPIcs.ICALP.2023.66, author = {Ghazi, Badih and Kamath, Pritish and Kumar, Ravi and Manurangsi, Pasin and Wu, Kewen}, title = {{On Differentially Private Counting on Trees}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {66:1--66:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.66}, URN = {urn:nbn:de:0030-drops-181186}, doi = {10.4230/LIPIcs.ICALP.2023.66}, annote = {Keywords: Differential Privacy, Algorithms, Trees, Hierarchies} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

We study the search problem class PPA_q defined as a modulo-q analog of the well-known polynomial parity argument class PPA introduced by Papadimitriou (JCSS 1994). Our first result shows that this class can be characterized in terms of PPA_p for prime p.
Our main result is to establish that an explicit version of a search problem associated to the Chevalley - Warning theorem is complete for PPA_p for prime p. This problem is natural in that it does not explicitly involve circuits as part of the input. It is the first such complete problem for PPA_p when p ≥ 3.
Finally we discuss connections between Chevalley-Warning theorem and the well-studied short integer solution problem and survey the structural properties of PPA_q.

Mika Göös, Pritish Kamath, Katerina Sotiraki, and Manolis Zampetakis. On the Complexity of Modulo-q Arguments and the Chevalley - Warning Theorem. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 19:1-19:42, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{goos_et_al:LIPIcs.CCC.2020.19, author = {G\"{o}\"{o}s, Mika and Kamath, Pritish and Sotiraki, Katerina and Zampetakis, Manolis}, title = {{On the Complexity of Modulo-q Arguments and the Chevalley - Warning Theorem}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {19:1--19:42}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.19}, URN = {urn:nbn:de:0030-drops-125712}, doi = {10.4230/LIPIcs.CCC.2020.19}, annote = {Keywords: Total NP Search Problems, Modulo-q arguments, Chevalley - Warning Theorem} }

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

Separations: We introduce a monotone variant of Xor-Sat and show it has exponential monotone circuit complexity. Since Xor-Sat is in NC^2, this improves qualitatively on the monotone vs. non-monotone separation of Tardos (1988). We also show that monotone span programs over R can be exponentially more powerful than over finite fields. These results can be interpreted as separating subclasses of TFNP in communication complexity.
Characterizations: We show that the communication (resp. query) analogue of PPA (subclass of TFNP) captures span programs over F_2 (resp. Nullstellensatz degree over F_2). Previously, it was known that communication FP captures formulas (Karchmer - Wigderson, 1988) and that communication PLS captures circuits (Razborov, 1995).

Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in Monotone Complexity and TFNP. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{goos_et_al:LIPIcs.ITCS.2019.38, author = {G\"{o}\"{o}s, Mika and Kamath, Pritish and Robere, Robert and Sokolov, Dmitry}, title = {{Adventures in Monotone Complexity and TFNP}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {38:1--38:19}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.38}, URN = {urn:nbn:de:0030-drops-101316}, doi = {10.4230/LIPIcs.ITCS.2019.38}, annote = {Keywords: TFNP, Monotone Complexity, Communication Complexity, Proof Complexity} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016].
Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest.

Badih Ghazi, Pritish Kamath, and Prasad Raghavendra. Dimension Reduction for Polynomials over Gaussian Space and Applications. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 28:1-28:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ghazi_et_al:LIPIcs.CCC.2018.28, author = {Ghazi, Badih and Kamath, Pritish and Raghavendra, Prasad}, title = {{Dimension Reduction for Polynomials over Gaussian Space and Applications}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {28:1--28:37}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.28}, URN = {urn:nbn:de:0030-drops-88616}, doi = {10.4230/LIPIcs.CCC.2018.28}, annote = {Keywords: Dimension reduction, Low-degree Polynomials, Noise Stability, Non-Interactive Simulation} }

Document

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

Motivated by an attempt to understand the formation and development of (human) language, we introduce a "distributed compression" problem. In our problem a sequence of pairs of players from a set of K players are chosen and tasked to communicate messages drawn from an unknown distribution Q.
Arguably languages are created and evolve to compress frequently occurring messages, and we focus on this aspect.
The only knowledge that players have about the distribution Q is from previously drawn samples, but these samples differ from player to player.
The only common knowledge between the players is restricted to a common prior distribution P and some constant number
of bits of information (such as a learning algorithm).
Letting T_epsilon denote the number of iterations it would take for a typical player
to obtain an epsilon-approximation to Q in total variation distance, we ask
whether T_epsilon iterations suffice to compress the messages down roughly to their
entropy and give a partial positive answer.
We show that a natural uniform algorithm can compress the communication down to an average cost per
message of O(H(Q) + log (D(P || Q)) in tilde{O}(T_epsilon) iterations
while allowing for O(epsilon)-error,
where D(. || .) denotes the KL-divergence between distributions.
For large divergences
this compares favorably with the static algorithm that ignores all samples and
compresses down to H(Q) + D(P || Q) bits, while not requiring T_epsilon * K iterations that it would take players to develop optimal but separate compressions for
each pair of players.
Along the way we introduce a "data-structural" view of the task of
communicating with a natural language and show that our natural algorithm can also be
implemented by an efficient data structure, whose storage is comparable to the storage requirements of Q and whose query complexity is comparable to the lengths of the message to be
compressed.
Our results give a plausible mathematical analogy to the mechanisms by which
human languages get created and evolve, and in particular highlights the
possibility of coordination towards a joint task (agreeing on a language)
while engaging in distributed learning.

Badih Ghazi, Elad Haramaty, Pritish Kamath, and Madhu Sudan. Compression in a Distributed Setting. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ghazi_et_al:LIPIcs.ITCS.2017.19, author = {Ghazi, Badih and Haramaty, Elad and Kamath, Pritish and Sudan, Madhu}, title = {{Compression in a Distributed Setting}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {19:1--19:22}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.19}, URN = {urn:nbn:de:0030-drops-81763}, doi = {10.4230/LIPIcs.ITCS.2017.19}, annote = {Keywords: Distributed Compression, Communication, Language Evolution, Isolating Hash Families} }

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**Published in:** LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

We prove that the P^NP-type query complexity (alternatively, decision list width) of any boolean function f is quadratically related to the P^NP-type communication complexity of a lifted version of f. As an application, we show that a certain "product" lower bound method of Impagliazzo and Williams (CCC 2010) fails to capture P^NP communication complexity up to polynomial factors, which answers a question of Papakonstantinou, Scheder, and Song (CCC 2014).

Mika Göös, Pritish Kamath, Toniann Pitassi, and Thomas Watson. Query-to-Communication Lifting for P^NP. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{goos_et_al:LIPIcs.CCC.2017.12, author = {G\"{o}\"{o}s, Mika and Kamath, Pritish and Pitassi, Toniann and Watson, Thomas}, title = {{Query-to-Communication Lifting for P^NP}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {12:1--12:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-040-8}, ISSN = {1868-8969}, year = {2017}, volume = {79}, editor = {O'Donnell, Ryan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.12}, URN = {urn:nbn:de:0030-drops-75388}, doi = {10.4230/LIPIcs.CCC.2017.12}, annote = {Keywords: Communication Complexity, Query Complexity, Lifting Theorem, P^NP} }

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

We introduce the notion of one-way communication schemes with partial noiseless feedback. In this setting, Alice wishes to communicate a message to Bob by using a communication scheme that involves sending a sequence of bits over a channel while receiving feedback bits from Bob for delta fraction of the transmissions. An adversary is allowed to corrupt up to a constant fraction of Alice's transmissions, while the feedback is always uncorrupted. Motivated by questions related to coding for interactive communication, we seek to determine the maximum error rate, as a function of 0 <= delta <= 1, such that Alice can send a message to Bob via some protocol with delta fraction of noiseless feedback. The case delta = 1 corresponds to full feedback, in which the result of Berlekamp ['64] implies that the maximum tolerable error rate is 1/3, while the case delta = 0 corresponds to no feedback, in which the maximum tolerable error rate is 1/4, achievable by use of a binary error-correcting code.
In this work, we show that for any delta in (0,1] and gamma in [0, 1/3), there exists a randomized communication scheme with noiseless delta-feedback, such that the probability of miscommunication is low, as long as no more than a gamma fraction of the rounds are corrupted. Moreover, we show that for any delta in (0, 1] and gamma < f(delta), there exists a deterministic communication scheme with noiseless delta-feedback that always decodes correctly as long as no more than a gamma fraction of rounds are corrupted. Here f is a monotonically increasing, piecewise linear, continuous function with f(0) = 1/4 and f(1) = 1/3. Also, the rate of communication in both cases is constant (dependent on delta and gamma but independent of the input length).

Bernhard Haeupler, Pritish Kamath, and Ameya Velingker. Communication with Partial Noiseless Feedback. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 881-897, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{haeupler_et_al:LIPIcs.APPROX-RANDOM.2015.881, author = {Haeupler, Bernhard and Kamath, Pritish and Velingker, Ameya}, title = {{Communication with Partial Noiseless Feedback}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {881--897}, 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.2015.881}, URN = {urn:nbn:de:0030-drops-53426}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.881}, annote = {Keywords: Communication with feedback, Interactive communication, Coding theory Digital} }

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**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

One central issue in the formal design and analysis of reactive systems is the notion of refinement that asks whether all behaviors of the implementation is allowed by the specification. The local interpretation of behavior leads to the notion of simulation. Alternating transition systems (ATSs) provide a general model for composite reactive systems, and the simulation relation for ATSs is known as alternating simulation. The simulation relation for fair transition systems is called fair simulation. In this work our main contributions are as follows: (1) We present an improved algorithm for fair simulation with Büchi fairness constraints; our algorithm requires O(n^3 * m) time as compared to the previous known O(n^6)-time algorithm, where n is the number of states and m is the number of transitions. (2) We present a game based algorithm for alternating simulation that requires O(m^2)-time as compared to the previous known O((n*m)^2)-time algorithm, where n is the number of states and m is the size of transition relation. (3) We present an iterative algorithm for alternating simulation that matches the time complexity of the game based algorithm, but is more space efficient than the game based algorithm.

Krishnendu Chatterjee, Siddhesh Chaubal, and Pritish Kamath. Faster Algorithms for Alternating Refinement Relations. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 167-182, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{chatterjee_et_al:LIPIcs.CSL.2012.167, author = {Chatterjee, Krishnendu and Chaubal, Siddhesh and Kamath, Pritish}, title = {{Faster Algorithms for Alternating Refinement Relations}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {167--182}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.167}, URN = {urn:nbn:de:0030-drops-36713}, doi = {10.4230/LIPIcs.CSL.2012.167}, annote = {Keywords: Simulation and fair simulation, Alternating simulation, Graph games} }

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