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**Published in:** LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

We initiate the study of error correcting codes over the multi-party adversarial broadcast channel. Specifically, we consider the classic information dissemination problem where n parties, each holding an input bit, wish to know each other’s input. For this, they communicate in rounds, where, in each round, one designated party sends a bit to all other parties over a channel governed by an adversary that may corrupt a constant fraction of the received communication. We mention that the dissemination problem was studied in the stochastic noise model since the 80’s.
While stochastic noise in multi-party channels has received quite a bit of attention, the case of adversarial noise has largely been avoided, as such channels cannot handle more than a 1/n-fraction of errors. Indeed, this many errors allow an adversary to completely corrupt the incoming or outgoing communication for one of the parties and fail the protocol. Curiously, we show that by eliminating these "trivial" attacks, one can get a simple protocol resilient to a constant fraction of errors. Thus, a model that rules out such attacks is both necessary and sufficient to get a resilient protocol.
The main shortcoming of our dissemination protocol is its length: it requires Θ(n²) communication rounds whereas n rounds suffice in the absence of noise. Our main result is a matching lower bound of Ω(n²) on the length of any dissemination protocol in our model. Our proof first "gets rid" of the channel noise by converting it to a form of "input noise", showing that a noisy dissemination protocol implies a (noiseless) protocol for a version of the direct sum gap-majority problem. We conclude the proof with a tight lower bound for the latter problem, which may be of independent interest.

Klim Efremenko, Gillat Kol, Dmitry Paramonov, Ran Raz, and Raghuvansh R. Saxena. Information Dissemination via Broadcasts in the Presence of Adversarial Noise. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 19:1-19:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{efremenko_et_al:LIPIcs.CCC.2024.19, author = {Efremenko, Klim and Kol, Gillat and Paramonov, Dmitry and Raz, Ran and Saxena, Raghuvansh R.}, title = {{Information Dissemination via Broadcasts in the Presence of Adversarial Noise}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {19:1--19:33}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.19}, URN = {urn:nbn:de:0030-drops-204159}, doi = {10.4230/LIPIcs.CCC.2024.19}, annote = {Keywords: Radio Networks, Interactive Coding, Error Correcting Codes} }

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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)

Single-hop radio networks (SHRN) are a well studied abstraction of communication over a wireless channel. In this model, in every round, each of the n participating parties may decide to broadcast a message to all the others, potentially causing collisions. We consider the SHRN model in the presence of stochastic message drops (i.e., erasures), where in every round, the message received by each party is erased (replaced by ⊥) with some small constant probability, independently.
Our main result is a constant rate coding scheme, allowing one to run protocols designed to work over the (noiseless) SHRN model over the SHRN model with erasures. Our scheme converts any protocol Π of length at most exponential in n over the SHRN model to a protocol Π' that is resilient to constant fraction of erasures and has length linear in the length of Π.
We mention that for the special case where the protocol Π is non-adaptive, i.e., the order of communication is fixed in advance, such a scheme was known. Nevertheless, adaptivity is widely used and is known to hugely boost the power of wireless channels, which makes handling the general case of adaptive protocols Π both important and more challenging. Indeed, to the best of our knowledge, our result is the first constant rate scheme that converts adaptive protocols to noise resilient ones in any multi-party model.

Klim Efremenko, Gillat Kol, Dmitry Paramonov, and Raghuvansh R. Saxena. Protecting Single-Hop Radio Networks from Message Drops. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 53:1-53:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{efremenko_et_al:LIPIcs.ICALP.2023.53, author = {Efremenko, Klim and Kol, Gillat and Paramonov, Dmitry and Saxena, Raghuvansh R.}, title = {{Protecting Single-Hop Radio Networks from Message Drops}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {53:1--53:20}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.53}, URN = {urn:nbn:de:0030-drops-181059}, doi = {10.4230/LIPIcs.ICALP.2023.53}, annote = {Keywords: Radio Networks, Interactive Coding, Error Correcting Codes} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Much of today’s communication is carried out over large wireless systems with different input-output behaviors. In this work, we compare the power of central abstractions of wireless communication through the general notion of boolean symmetric f-channels: In every round of the f-channel, each of its n parties decides to either broadcast or not, and the channel outputs f(m), where m is the number of broadcasting parties.
Our first result is that the well studied beeping channel, where f is the threshold-1 function, is not stronger than any other f-channel. To this end, we design a protocol over the f-channel and prove that any protocol that simulates it over the beeping channel blows up the round complexity by a factor of Ω(log n). Our lower bound technique may be of independent interest, as it essentially generalizes the popular fooling set technique by exploiting a "local" relaxation of combinatorial rectangles.
Curiously, while this result shows the limitations of a noiseless channel, namely, the beeping channel, we are able to use it to show the limitations of the noisy version of many other channels. This includes the extensively studied single-hop radio network model with collisions-as-silence (CAS), which is equivalent to the f-channel with f(m) = 1 iff m = 1.
In particular, our second and main result, obtained from the first, shows that converting CAS protocols to noise resilient ones may incur a large performance overhead, i.e., no constant rate interactive code exists. To this end, we design a CAS protocol and prove that any protocol that simulates it over the noisy CAS model with correlated stochastic noise, blows up the round complexity by a factor of Ω(log n). We mention that the Ω(log n) overhead in both our results is tight.

Klim Efremenko, Gillat Kol, Dmitry Paramonov, and Raghuvansh R. Saxena. Noisy Radio Network Lower Bounds via Noiseless Beeping Lower Bounds. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{efremenko_et_al:LIPIcs.ITCS.2023.46, author = {Efremenko, Klim and Kol, Gillat and Paramonov, Dmitry and Saxena, Raghuvansh R.}, title = {{Noisy Radio Network Lower Bounds via Noiseless Beeping Lower Bounds}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {46:1--46:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.46}, URN = {urn:nbn:de:0030-drops-175499}, doi = {10.4230/LIPIcs.ITCS.2023.46}, annote = {Keywords: Beeping Model, Radio Networks} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We study boolean constraint satisfaction problems (CSPs) Max-CSP^f_n for all predicates f: {0,1}^k → {0,1}. In these problems, given an integer v and a list of constraints over n boolean variables, each obtained by applying f to a sequence of literals, we wish to decide if there is an assignment to the variables that satisfies at least v constraints. We consider these problems in the streaming model, where the algorithm makes a small number of passes over the list of constraints.
Our first and main result is the following complete characterization: For every predicate f, the streaming space complexity of the Max-CSP^f_n problem is Θ̃(n^deg(f)), where deg(f) is the degree of f when viewed as a multilinear polynomial. While the upper bound is obtained by a (very simple) one-pass streaming algorithm, our lower bound shows that a better space complexity is impossible even with constant-pass streaming algorithms.
Building on our techniques, we are also able to get an optimal Ω(n²) lower bound on the space complexity of constant-pass streaming algorithms for the well studied Max-CUT problem, even though it is not technically a Max-CSP^f_n problem as, e.g., negations of variables and repeated constraints are not allowed.

Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, and Huacheng Yu. Characterizing the Multi-Pass Streaming Complexity for Solving Boolean CSPs Exactly. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 80:1-80:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kol_et_al:LIPIcs.ITCS.2023.80, author = {Kol, Gillat and Paramonov, Dmitry and Saxena, Raghuvansh R. and Yu, Huacheng}, title = {{Characterizing the Multi-Pass Streaming Complexity for Solving Boolean CSPs Exactly}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {80:1--80:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.80}, URN = {urn:nbn:de:0030-drops-175837}, doi = {10.4230/LIPIcs.ITCS.2023.80}, annote = {Keywords: Streaming algorithms, Constraint Satisfaction Problems} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We consider prophet inequalities subject to feasibility constraints that are the intersection of q matroids. The best-known algorithms achieve a Θ(q)-approximation, even when restricted to instances that are the intersection of q partition matroids, and with i.i.d. Bernoulli random variables [José R. Correa et al., 2022; Moran Feldman et al., 2016; Marek Adamczyk and Michal Wlodarczyk, 2018]. The previous best-known lower bound is Θ(√q) due to a simple construction of [Robert Kleinberg and S. Matthew Weinberg, 2012] (which uses i.i.d. Bernoulli random variables, and writes the construction as the intersection of partition matroids).
We establish an improved lower bound of q^{1/2+Ω(1/log log q)} by writing the construction of [Robert Kleinberg and S. Matthew Weinberg, 2012] as the intersection of asymptotically fewer partition matroids. We accomplish this via an improved upper bound on the product dimension of a graph with p^p disjoint cliques of size p, using recent techniques developed in [Noga Alon and Ryan Alweiss, 2020].

Raghuvansh R. Saxena, Santhoshini Velusamy, and S. Matthew Weinberg. An Improved Lower Bound for Matroid Intersection Prophet Inequalities. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 95:1-95:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{saxena_et_al:LIPIcs.ITCS.2023.95, author = {Saxena, Raghuvansh R. and Velusamy, Santhoshini and Weinberg, S. Matthew}, title = {{An Improved Lower Bound for Matroid Intersection Prophet Inequalities}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {95:1--95:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.95}, URN = {urn:nbn:de:0030-drops-175986}, doi = {10.4230/LIPIcs.ITCS.2023.95}, annote = {Keywords: Prophet Inequalities, Intersection of Matroids} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

For a directed graph G with n vertices and a start vertex u_start, we wish to (approximately) sample an L-step random walk over G starting from u_start with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be Θ̃(n ⋅ L). Prior to our work, a better space complexity was only known with Õ(√L) passes.
We essentially settle the space complexity of this random walk simulation problem for two-pass streaming algorithms, showing that it is Θ̃(n ⋅ √L), by giving almost matching upper and lower bounds. Our lower bound argument extends to every constant number of passes p, and shows that any p-pass algorithm for this problem uses Ω̃(n ⋅ L^{1/p}) space. In addition, we show a similar Θ̃(n ⋅ √L) bound on the space complexity of any algorithm (with any number of passes) for the related problem of sampling an L-step random walk from every vertex in the graph.

Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, and Huacheng Yu. Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 52:1-52:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2021.52, author = {Chen, Lijie and Kol, Gillat and Paramonov, Dmitry and Saxena, Raghuvansh R. and Song, Zhao and Yu, Huacheng}, title = {{Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {52:1--52:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.52}, URN = {urn:nbn:de:0030-drops-141218}, doi = {10.4230/LIPIcs.ICALP.2021.52}, annote = {Keywords: streaming algorithms, random walk sampling} }

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

We study the n-party noisy broadcast channel with a constant fraction of malicious parties. Specifically, we assume that each non-malicious party holds an input bit, and communicates with the others in order to learn the input bits of all non-malicious parties. In each communication round, one of the parties broadcasts a bit to all other parties, and the bit received by each party is flipped with a fixed constant probability (independently for each recipient). How many rounds are needed?
Assuming there are no malicious parties, Gallager gave an 𝒪(n log log n)-round protocol for the above problem, which was later shown to be optimal. This protocol, however, inherently breaks down in the presence of malicious parties.
We present a novel n ⋅ 𝒪̃(√{log n})-round protocol, that solves this problem even when almost half of the parties are malicious. Our protocol uses a new type of error correcting code, which we call a locality sensitive code and which may be of independent interest. Roughly speaking, these codes map "close" messages to "close" codewords, while messages that are not close are mapped to codewords that are very far apart.
We view our result as a first step towards a theory of property preserving interactive coding, i.e., interactive codes that preserve useful properties of the protocol being encoded. In our case, the naive protocol over the noiseless broadcast channel, where all the parties broadcast their input bit and output all the bits received, works even in the presence of malicious parties. Our simulation of this protocol, unlike Gallager’s, preserves this property of the original protocol.

Klim Efremenko, Gillat Kol, Dmitry Paramonov, and Raghuvansh R. Saxena. Computation over the Noisy Broadcast Channel with Malicious Parties. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 82:1-82:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{efremenko_et_al:LIPIcs.ITCS.2021.82, author = {Efremenko, Klim and Kol, Gillat and Paramonov, Dmitry and Saxena, Raghuvansh R.}, title = {{Computation over the Noisy Broadcast Channel with Malicious Parties}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {82:1--82:19}, 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.2021.82}, URN = {urn:nbn:de:0030-drops-136215}, doi = {10.4230/LIPIcs.ITCS.2021.82}, annote = {Keywords: Broadcast Network, Malicious Parties, Communication Complexity} }