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APPROX

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every k ≥ 5, we show that CSPs where the underlying predicate is a pure monarchy function on k variables have no non-trivial sketching approximation algorithm in o(√n) space. We also show infinitely many weaker monarchy functions for which CSPs using such constraints are non-trivially approximable by O(log(n)) space sketching algorithms. Moreover, we give the first example of sketching approximable asymmetric Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan, and Velusamy (FOCS 2021) that characterizes the sketching approximability of all CSPs. Their framework can be applied naturally to get a computer-aided analysis of the approximability of any specific constraint satisfaction problem. The novelty of our work is in using their work to get an analysis that applies to infinitely many problems simultaneously.

Chi-Ning Chou, Alexander Golovnev, Amirbehshad Shahrasbi, Madhu Sudan, and Santhoshini Velusamy. Sketching Approximability of (Weak) Monarchy Predicates. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 35:1-35:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chou_et_al:LIPIcs.APPROX/RANDOM.2022.35, author = {Chou, Chi-Ning and Golovnev, Alexander and Shahrasbi, Amirbehshad and Sudan, Madhu and Velusamy, Santhoshini}, title = {{Sketching Approximability of (Weak) Monarchy Predicates}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {35:1--35:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.35}, URN = {urn:nbn:de:0030-drops-171573}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.35}, annote = {Keywords: sketching algorithms, approximability, linear threshold functions} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We present many new results related to reliable (interactive) communication over insertion-deletion channels. Synchronization errors, such as insertions and deletions, strictly generalize the usual symbol corruption errors and are much harder to protect against.
We show how to hide the complications of synchronization errors in many applications by introducing very general channel simulations which efficiently transform an insertion-deletion channel into a regular symbol corruption channel with an error rate larger by a constant factor and a slightly smaller alphabet. We utilize and generalize synchronization string based methods which were recently introduced as a tool to design essentially optimal error correcting codes for insertion-deletion channels. Our channel simulations depend on the fact that, at the cost of increasing the error rate by a constant factor, synchronization strings can be decoded in a streaming manner that preserves linearity of time. Interestingly, we provide a lower bound showing that this constant factor cannot be improved to 1+epsilon, in contrast to what is achievable for error correcting codes. Our channel simulations drastically and cleanly generalize the applicability of synchronization strings.
We provide new interactive coding schemes which simulate any interactive two-party protocol over an insertion-deletion channel. Our results improve over the interactive coding schemes of Braverman et al. [TransInf `17] and Sherstov and Wu [FOCS `17] which achieve a small constant rate and require exponential time computations with respect to computational and communication complexities. We provide the first computationally efficient interactive coding schemes for synchronization errors, the first coding scheme with a rate approaching one for small noise rates, and also the first coding scheme that works over arbitrarily small alphabet sizes. We also show tight connections between synchronization strings and edit-distance tree codes which allow us to transfer results from tree codes directly to edit-distance tree codes.
Finally, using on our channel simulations, we provide an explicit low-rate binary insertion-deletion code that improves over the state-of-the-art codes by Guruswami and Wang [TransInf `17] in terms of rate-distance trade-off.

Bernhard Haeupler, Amirbehshad Shahrasbi, and Ellen Vitercik. Synchronization Strings: Channel Simulations and Interactive Coding for Insertions and Deletions. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 75:1-75:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{haeupler_et_al:LIPIcs.ICALP.2018.75, author = {Haeupler, Bernhard and Shahrasbi, Amirbehshad and Vitercik, Ellen}, title = {{Synchronization Strings: Channel Simulations and Interactive Coding for Insertions and Deletions}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {75:1--75:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.75}, URN = {urn:nbn:de:0030-drops-90794}, doi = {10.4230/LIPIcs.ICALP.2018.75}, annote = {Keywords: Synchronization Strings, Channel Simulation, Coding for Interactive Communication} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We study codes that are list-decodable under insertions and deletions ("insdel codes"). Specifically, we consider the setting where, given a codeword x of length n over some finite alphabet Sigma of size q, delta * n codeword symbols may be adversarially deleted and gamma * n symbols may be adversarially inserted to yield a corrupted word w. A code is said to be list-decodable if there is an (efficient) algorithm that, given w, reports a small list of codewords that include the original codeword x. Given delta and gamma we study what is the rate R for which there exists a constant q and list size L such that there exist codes of rate R correcting delta-fraction insertions and gamma-fraction deletions while reporting lists of size at most L.
Using the concept of synchronization strings, introduced by the first two authors [Proc. STOC 2017], we show some surprising results. We show that for every 0 <= delta < 1, every 0 <= gamma < infty and every epsilon > 0 there exist codes of rate 1 - delta - epsilon and constant alphabet (so q = O_{delta,gamma,epsilon}(1)) and sub-logarithmic list sizes. Furthermore, our codes are accompanied by efficient (polynomial time) decoding algorithms. We stress that the fraction of insertions can be arbitrarily large (more than 100%), and the rate is independent of this parameter. We also prove several tight bounds on the parameters of list-decodable insdel codes. In particular, we show that the alphabet size of insdel codes needs to be exponentially large in epsilon^{-1}, where epsilon is the gap to capacity above. Our result even applies to settings where the unique-decoding capacity equals the list-decoding capacity and when it does so, it shows that the alphabet size needs to be exponentially large in the gap to capacity. This is sharp contrast to the Hamming error model where alphabet size polynomial in epsilon^{-1} suffices for unique decoding. This lower bound also shows that the exponential dependence on the alphabet size in previous works that constructed insdel codes is actually necessary!
Our result sheds light on the remarkable asymmetry between the impact of insertions and deletions from the point of view of error-correction: Whereas deletions cost in the rate of the code, insertion costs are borne by the adversary and not the code! Our results also highlight the dominance of the model of insertions and deletions over the Hamming model: A Hamming error is equal to one insertion and one deletion (at the same location). Thus the effect of delta-fraction Hamming errors can be simulated by delta-fraction of deletions and delta-fraction of insertions - but insdel codes can deal with much more insertions without loss in rate (though at the price of higher alphabet size).

Bernhard Haeupler, Amirbehshad Shahrasbi, and Madhu Sudan. Synchronization Strings: List Decoding for Insertions and Deletions. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 76:1-76:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{haeupler_et_al:LIPIcs.ICALP.2018.76, author = {Haeupler, Bernhard and Shahrasbi, Amirbehshad and Sudan, Madhu}, title = {{Synchronization Strings: List Decoding for Insertions and Deletions}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {76:1--76:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.76}, URN = {urn:nbn:de:0030-drops-90807}, doi = {10.4230/LIPIcs.ICALP.2018.76}, annote = {Keywords: List Decoding, Insertions and Deletions, Synchronization Strings} }

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