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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)

Given a function f on F₂ⁿ, we study the following problem. What is the largest affine subspace 𝒰 such that when restricted to 𝒰, all the non-trivial Fourier coefficients of f are very small?
For the natural class of bounded Fourier degree d functions f: F₂ⁿ → [-1,1], we show that there exists an affine subspace of dimension at least Ω(n^{1/d!} k^{-2}), wherein all of f’s nontrivial Fourier coefficients become smaller than 2^{-k}. To complement this result, we show the existence of degree d functions with coefficients larger than 2^{-d log n} when restricted to any affine subspace of dimension larger than Ω(d n^{1/(d-1)}). In addition, we give explicit examples of functions with analogous but weaker properties.
Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of F₂ⁿ that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers.

Siddharth Iyer and Michael Whitmeyer. Searching for Regularity in Bounded Functions. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 83:1-83:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{iyer_et_al:LIPIcs.ICALP.2023.83, author = {Iyer, Siddharth and Whitmeyer, Michael}, title = {{Searching for Regularity in Bounded Functions}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {83:1--83:20}, 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.2023.83}, URN = {urn:nbn:de:0030-drops-181351}, doi = {10.4230/LIPIcs.ICALP.2023.83}, annote = {Keywords: regularity, bounded function, Boolean function, Fourier analysis} }

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**Published in:** LIPIcs, Volume 153, 23rd International Conference on Principles of Distributed Systems (OPODIS 2019)

We consider distributed computations between two parties carried out over a noisy channel that may erase messages. Following a noise model proposed by Dani et al. (2018), the noise level observed by the parties during the computation in our setting is arbitrary and a priori unknown to the parties.
We develop interactive coding schemes that adapt to the actual level of noise and correctly execute any two-party computation. Namely, in case the channel erases T transmissions, the coding scheme will take N+2T transmissions using an alphabet of size 4 (alternatively, using 2N+4T transmissions over a binary channel) to correctly simulate any binary protocol that takes N transmissions assuming a noiseless channel. We can further reduce the communication to N+T by relaxing the communication model and allowing parties to remain silent rather than forcing them to communicate in every round of the coding scheme.
Our coding schemes are efficient, deterministic, have linear overhead both in their communication and round complexity, and succeed (with probability 1) regardless of the number of erasures T.

Ran Gelles and Siddharth Iyer. Interactive Coding Resilient to an Unknown Number of Erasures. In 23rd International Conference on Principles of Distributed Systems (OPODIS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 153, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gelles_et_al:LIPIcs.OPODIS.2019.13, author = {Gelles, Ran and Iyer, Siddharth}, title = {{Interactive Coding Resilient to an Unknown Number of Erasures}}, booktitle = {23rd International Conference on Principles of Distributed Systems (OPODIS 2019)}, pages = {13:1--13:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-133-7}, ISSN = {1868-8969}, year = {2020}, volume = {153}, editor = {Felber, Pascal and Friedman, Roy and Gilbert, Seth and Miller, Avery}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2019.13}, URN = {urn:nbn:de:0030-drops-117999}, doi = {10.4230/LIPIcs.OPODIS.2019.13}, annote = {Keywords: Interactive coding, erasure channels, distributed computation with noise, unbounded noise} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

The well-known k-disjoint path problem (k-DPP) asks for pairwise vertex-disjoint paths between k specified pairs of vertices (s_i, t_i) in a given graph, if they exist. The decision version of the shortest k-DPP asks for the length of the shortest (in terms of total length) such paths. Similarly, the search and counting versions ask for one such and the number of such shortest set of paths, respectively.
We restrict attention to the shortest k-DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequential and parallel algorithms for the search versions of the problem answering one of the main open questions raised by Colin de Verdière and Schrijver [Éric Colin de Verdière and Alexander Schrijver, 2011] for the general one-face problem. We do so by providing a randomised NC^2 algorithm along with an O(n^{omega/2}) time randomised sequential algorithm, for any fixed k. We also obtain deterministic algorithms with similar resource bounds for the counting and search versions. In contrast, previously, only the sequential complexity of decision and search versions of the "well-ordered" case has been studied. For the one-face case, sequential versions of our routines have better running times for constantly many terminals.
The algorithms are based on a bijection between a shortest k-tuple of disjoint paths in the given graph and cycle covers in a related digraph. This allows us to non-trivially modify established techniques relating counting cycle covers to the determinant. We further need to do a controlled inclusion-exclusion to produce a polynomial sum of determinants such that all "bad" cycle covers cancel out in the sum allowing us to count "pure" cycle covers.

Samir Datta, Siddharth Iyer, Raghav Kulkarni, and Anish Mukherjee. Shortest k-Disjoint Paths via Determinants. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 19:1-19:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{datta_et_al:LIPIcs.FSTTCS.2018.19, author = {Datta, Samir and Iyer, Siddharth and Kulkarni, Raghav and Mukherjee, Anish}, title = {{Shortest k-Disjoint Paths via Determinants}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {19:1--19:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.19}, URN = {urn:nbn:de:0030-drops-99183}, doi = {10.4230/LIPIcs.FSTTCS.2018.19}, annote = {Keywords: disjoint paths, planar graph, parallel algorithm, cycle cover, determinant, inclusion-exclusion} }