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RANDOM

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

A tournament is a complete directed graph. A source in a tournament is a vertex that has no in-neighbours (every other vertex is reachable from it via a path of length 1), and a king in a tournament is a vertex v such that every other vertex is reachable from v via a path of length at most 2. It is well known that every tournament has at least one king. In particular, a maximum out-degree vertex is a king. The tasks of finding a king and a maximum out-degree vertex in a tournament has been relatively well studied in the context of query complexity. We study the communication complexity of finding a king, of finding a maximum out-degree vertex, and of finding a source (if it exists) in a tournament, where the edges are partitioned between two players. The following are our main results for n-vertex tournaments:
- We show that the communication task of finding a source in a tournament is equivalent to the well-studied Clique vs. Independent Set (CIS) problem on undirected graphs. As a result, known bounds on the communication complexity of CIS [Yannakakis, JCSS'91, Göös, Pitassi, Watson, SICOMP'18] imply a bound of Θ̃(log² n) for finding a source (if it exists, or outputting that there is no source) in a tournament.
- The deterministic and randomized communication complexities of finding a king are Θ(n). The quantum communication complexity of finding a king is Θ̃(√n).
- The deterministic, randomized, and quantum communication complexities of finding a maximum out-degree vertex are Θ(n log n), Θ̃(n) and Θ̃(√n), respectively. Our upper bounds above hold for all partitions of edges, and the lower bounds for a specific partition of the edges.
One of our lower bounds uses a fooling-set based argument, and all our other lower bounds follow from carefully-constructed reductions from Set-Disjointness. An interesting point to note here is that while the deterministic query complexity of finding a king has been open for over two decades [Shen, Sheng, Wu, SICOMP'03], we are able to essentially resolve the complexity of this problem in a model (communication complexity) that is usually harder to analyze than query complexity.

Nikhil S. Mande, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the Communication Complexity of Finding a King in a Tournament. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 64:1-64:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mande_et_al:LIPIcs.APPROX/RANDOM.2024.64, author = {Mande, Nikhil S. and Paraashar, Manaswi and Sanyal, Swagato and Saurabh, Nitin}, title = {{On the Communication Complexity of Finding a King in a Tournament}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {64:1--64:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.64}, URN = {urn:nbn:de:0030-drops-210571}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.64}, annote = {Keywords: Communication complexity, tournaments, query complexity} }

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RANDOM

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e. functions obtained by composing a base function with itself a number of times. Let h^d denote the standard d-fold composition of the base function h. The main result of this work is to show that the approximate degree composes if either of the following conditions holds:
- The outer function f:{0,1}ⁿ → {0,1} is a recursive function of the form h^d, with h being any base function and d = Ω(log log n).
- The inner function is a recursive function of the form h^d, with h being any constant arity base function (other than AND and OR) and d = Ω(log log n), where n is the arity of the outer function.
In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be efficiently eliminated if the inner or outer function is a recursive function.

Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, and Nitin Saurabh. Approximate Degree Composition for Recursive Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 71:1-71:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chakraborty_et_al:LIPIcs.APPROX/RANDOM.2024.71, author = {Chakraborty, Sourav and Kayal, Chandrima and Mittal, Rajat and Paraashar, Manaswi and Saurabh, Nitin}, title = {{Approximate Degree Composition for Recursive Functions}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {71:1--71:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.71}, URN = {urn:nbn:de:0030-drops-210642}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.71}, annote = {Keywords: Approximate degree, Boolean function, Composition theorem} }

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**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

A tournament is a complete directed graph. It is well known that every tournament contains at least one vertex v such that every other vertex is reachable from v by a path of length at most 2. All such vertices v are called kings of the underlying tournament. Despite active recent research in the area, the best-known upper and lower bounds on the deterministic query complexity (with query access to directions of edges) of finding a king in a tournament on n vertices are from over 20 years ago, and the bounds do not match: the best-known lower bound is Ω(n^{4/3}) and the best-known upper bound is O(n^{3/2}) [Shen, Sheng, Wu, SICOMP'03]. Our contribution is to show tight bounds (up to logarithmic factors) of Θ̃(n) and Θ̃(√n) in the randomized and quantum query models, respectively. We also study the randomized and quantum query complexities of finding a maximum out-degree vertex in a tournament.

Nikhil S. Mande, Manaswi Paraashar, and Nitin Saurabh. Randomized and Quantum Query Complexities of Finding a King in a Tournament. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{mande_et_al:LIPIcs.FSTTCS.2023.30, author = {Mande, Nikhil S. and Paraashar, Manaswi and Saurabh, Nitin}, title = {{Randomized and Quantum Query Complexities of Finding a King in a Tournament}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {30:1--30:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.30}, URN = {urn:nbn:de:0030-drops-194039}, doi = {10.4230/LIPIcs.FSTTCS.2023.30}, annote = {Keywords: Query complexity, quantum computing, randomized query complexity, tournament solutions, search problems} }

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**Published in:** Dagstuhl Reports, Volume 13, Issue 3 (2023)

This report documents the program and activities of Dagstuhl Seminar 23111 "Computational Complexity of Discrete Problems", which was held in-person in March 2023 (the previous instance of the seminar series had been held online in March 2021). Following a description of the seminar’s objectives and its overall organization, this report lists the different major talks given during the seminar in alphabetical order of speakers, followed by the abstracts of the talks, including the main references and relevant sources where applicable. The return to an in-person setting allowed an intense atmosphere of active research and interaction throughout the five day seminar.

Anna Gál, Meena Mahajan, Rahul Santhanam, Till Tantau, and Manaswi Paraashar. Computational Complexity of Discrete Problems (Dagstuhl Seminar 23111). In Dagstuhl Reports, Volume 13, Issue 3, pp. 17-31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{gal_et_al:DagRep.13.3.17, author = {G\'{a}l, Anna and Mahajan, Meena and Santhanam, Rahul and Tantau, Till and Paraashar, Manaswi}, title = {{Computational Complexity of Discrete Problems (Dagstuhl Seminar 23111)}}, pages = {17--31}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {13}, number = {3}, editor = {G\'{a}l, Anna and Mahajan, Meena and Santhanam, Rahul and Tantau, Till and Paraashar, Manaswi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.13.3.17}, URN = {urn:nbn:de:0030-drops-192261}, doi = {10.4230/DagRep.13.3.17}, annote = {Keywords: circuit complexity, communication complexity, computational complexity, lower bounds, randomness} }

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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

For any Boolean functions f and g, the question whether R(f∘g) = Θ̃(R(f) ⋅ R(g)), is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg̃(f∘g) = Θ̃(deg̃(f)⋅deg̃(g)). These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily.
It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which R and deg̃ compose.
A recent landmark result (Ben-David and Blais, 2020) showed that R(f∘g) = Ω(noisyR(f)⋅ R(g)). This implies that composition holds whenever noisyR(f) = Θ̃(R(f)). We show two results:
1. When R(f) = Θ(n), then noisyR(f) = Θ(R(f)). In other words, composition holds whenever the randomized query complexity of the outer function is full.
2. If R composes with respect to an outer function, then noisyR also composes with respect to the same outer function. On the other hand, no result of the type deg̃(f∘g) = Ω(M(f) ⋅ deg̃(g)) (for some non-trivial complexity measure M(⋅)) was known to the best of our knowledge. We prove that deg̃(f∘g) = Ω̃(√{bs(f)} ⋅ deg̃(g)), where bs(f) is the block sensitivity of f. This implies that deg̃ composes when deg̃(f) is asymptotically equal to √{bs(f)}.
It is already known that both R and deg̃ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.

Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the Composition of Randomized Query Complexity and Approximate Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 63:1-63:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chakraborty_et_al:LIPIcs.APPROX/RANDOM.2023.63, author = {Chakraborty, Sourav and Kayal, Chandrima and Mittal, Rajat and Paraashar, Manaswi and Sanyal, Swagato and Saurabh, Nitin}, title = {{On the Composition of Randomized Query Complexity and Approximate Degree}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {63:1--63:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.63}, URN = {urn:nbn:de:0030-drops-188883}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.63}, annote = {Keywords: Approximate degree, Boolean functions, Composition Theorem, Partial functions, Randomized Query Complexity} }

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**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

For an unknown n × n matrix A having non-negative entries, the inner product (IP) oracle takes as inputs a specified row (or a column) of A and a vector 𝐯 ∈ ℝⁿ with non-negative entries, and returns their inner product. Given two input vectors x and y in ℝⁿ with non-negative entries, and an unknown matrix A with non-negative entries with IP oracle access, we design almost optimal sublinear time algorithms for the following two fundamental matrix problems:
- Find an estimate 𝒳 for the bilinear form x^T A y such that 𝒳 ≈ x^T A y.
- Designing a sampler 𝒵 for the entries of the matrix A such that ℙ(𝒵 = (i,j)) ≈ x_i A_{ij} y_j /(x^T A y), where x_i and y_j are i-th and j-th coordinate of 𝐱 and 𝐲 respectively. As special cases of the above results, for any submatrix of an unknown matrix with non-negative entries and IP oracle access, we can efficiently estimate the sum of the entries of any submatrix, and also sample a random entry from the submatrix with probability proportional to its weight. We will show that the above results imply that if we are given IP oracle access to the adjacency matrix of a graph, with non-negative weights on the edges, then we can design sublinear time algorithms for the following two fundamental graph problems:
- Estimating the sum of the weights of the edges of an induced subgraph, and
- Sampling edges proportional to their weights from an induced subgraph. We show that compared to the classical local queries (degree, adjacency, and neighbor queries) on graphs, we can get a quadratic speedup if we use IP oracle access for the above two problems.
Apart from the above, we study several matrix problems through the lens of IP oracle, like testing if the matrix is diagonal, symmetric, doubly stochastic, etc. Note that IP oracle is in the class of linear algebraic queries used lately in a series of works by Ben-Eliezer et al. [SODA'08], Nisan [SODA'21], Rashtchian et al. [RANDOM'20], Sun et al. [ICALP'19], and Shi and Woodruff [AAAI'19]. Recently, IP oracle was used by Bishnu et al. [RANDOM'21] to estimate dissimilarities between two matrices.

Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar. Counting and Sampling from Substructures Using Linear Algebraic Queries. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bishnu_et_al:LIPIcs.FSTTCS.2022.8, author = {Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath and Paraashar, Manaswi}, title = {{Counting and Sampling from Substructures Using Linear Algebraic Queries}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {8:1--8:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.8}, URN = {urn:nbn:de:0030-drops-174009}, doi = {10.4230/LIPIcs.FSTTCS.2022.8}, annote = {Keywords: Query complexity, Bilinear form, Uniform sampling, Weighted graphs} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

The role of symmetry in Boolean functions f:{0, 1}ⁿ → {0, 1} has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of 𝖲_n, is an important class of functions in the study of Boolean functions. A function f:{0, 1}ⁿ → {0, 1} is called transitive (or weakly-symmetric) if there exists a transitive group 𝖦 of 𝖲_n such that f is invariant under the action of 𝖦. In other words, the value of the function remains unchanged even after the input bits of f are moved around according to some permutation σ ∈ 𝖦. Understanding various complexity measures of transitive functions has been a rich area of research for the past few decades.
This work studies transitive functions in light of several combinatorial measures. The question that we try to address in this paper is what are the maximum separations between various pairs of combinatorial measures for transitive functions. Such study for general Boolean functions has been going on for many years. Aaronson et al. (STOC, 2021) have nicely compiled the current best-known results for general Boolean functions. But before this paper, no such systematic study had been done on the case of transitive functions.
Separations between a pair of combinatorial measures are shown by constructing interesting functions that demonstrate the separation. Over the past three decades, various interesting classes of functions have been designed for this purpose. In this context, one of the celebrated classes of functions is the "pointer functions". Ambainis et al. (JACM, 2017) constructed several functions, which are modifications of the pointer function in Göös et al. (SICOMP, 2018 / FOCS, 2015), to demonstrate the separation between various pairs of measures. In the last few years, pointer functions have been used to show separation between various other pairs of measures (Eg: Mukhopadhyay et al. (FSTTCS, 2015), Ben-David et al. (ITCS, 2017), Göös et al. (ToCT, 2018 / ICALP, 2017)).
However, the pointer functions themselves are not transitive. Based on the various kinds of pointer functions, we construct new transitive functions, which we use to demonstrate similar separations between various pairs of combinatorial measures as demonstrated by the original pointer functions. Our construction of transitive functions depends crucially on the construction of particular classes of transitive groups whose actions, though involved, help to preserve certain structural features of the input strings. The transitive groups we construct may be of independent interest in other areas of mathematics and theoretical computer science.
We summarize the current knowledge of relations between various combinatorial measures of transitive functions in a table similar to the table compiled by Aaronson et al. (STOC, 2021) for general functions.

Sourav Chakraborty, Chandrima Kayal, and Manaswi Paraashar. Separations Between Combinatorial Measures for Transitive Functions. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakraborty_et_al:LIPIcs.ICALP.2022.36, author = {Chakraborty, Sourav and Kayal, Chandrima and Paraashar, Manaswi}, title = {{Separations Between Combinatorial Measures for Transitive Functions}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {36:1--36:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.36}, URN = {urn:nbn:de:0030-drops-163779}, doi = {10.4230/LIPIcs.ICALP.2022.36}, annote = {Keywords: Transitive functions, Combinatorial complexity of Boolean functions} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}ⁿ → {-1,1} and G ∈ {AND₂, XOR₂}, the bounded-error quantum communication complexity of the composed function f∘G equals O(𝖰(f) log n), where 𝖰(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that 𝖱^{cc}(f∘G) ≤ 2𝖱(f), where 𝖱^{cc} and 𝖱 denote bounded-error communication and query complexity, respectively. Chakraborty et al. (CCC'20) exhibited a total function for which the log n overhead in the BCW simulation is required. This established the somewhat surprising fact that quantum reductions are in some cases inherently more expensive than classical reductions. We improve upon their result in several ways.
- We show that the log n overhead is not required when f is symmetric (i.e., depends only on the Hamming weight of its input), generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). Our upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication.
- In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. This also provides a different, and arguably simpler, proof of Aaronson and Ambainis’s O(√n) communication upper bound for Set-Disjointness.
- In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2 (this corresponds to the unbounded-error model of communication).
- We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free.

Sourav Chakraborty, Arkadev Chattopadhyay, Peter Høyer, Nikhil S. Mande, Manaswi Paraashar, and Ronald de Wolf. Symmetry and Quantum Query-To-Communication Simulation. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakraborty_et_al:LIPIcs.STACS.2022.20, author = {Chakraborty, Sourav and Chattopadhyay, Arkadev and H{\o}yer, Peter and Mande, Nikhil S. and Paraashar, Manaswi and de Wolf, Ronald}, title = {{Symmetry and Quantum Query-To-Communication Simulation}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {20:1--20:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.20}, URN = {urn:nbn:de:0030-drops-158309}, doi = {10.4230/LIPIcs.STACS.2022.20}, annote = {Keywords: Classical and quantum communication complexity, query-to-communication-simulation, quantum computing} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Chang’s lemma (Duke Mathematical Journal, 2002) is a classical result in mathematics, with applications spanning across additive combinatorics, combinatorial number theory, analysis of Boolean functions, communication complexity and algorithm design. For a Boolean function f that takes values in {-1, 1} let r(f) denote its Fourier rank (i.e., the dimension of the span of its Fourier support). For each positive threshold t, Chang’s lemma provides a lower bound on δ(f) := Pr[f(x) = -1] in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least 1/t. In this work we examine the tightness of Chang’s lemma with respect to the following three natural settings of the threshold:
- the Fourier sparsity of f, denoted k(f),
- the Fourier max-supp-entropy of f, denoted k'(f), defined to be the maximum value of the reciprocal of the absolute value of a non-zero Fourier coefficient,
- the Fourier max-rank-entropy of f, denoted k''(f), defined to be the minimum t such that characters whose coefficients are at least 1/t in magnitude span a r(f)-dimensional space. In this work we prove new lower bounds on δ(f) in terms of the above measures. One of our lower bounds, δ(f) = Ω(r(f)²/(k(f) log² k(f))), subsumes and refines the previously best known upper bound r(f) = O(√{k(f)}log k(f)) on r(f) in terms of k(f) by Sanyal (Theory of Computing, 2019). We improve upon this bound and show r(f) = O(√{k(f)δ(f)}log k(f)). Another lower bound, δ(f) = Ω(r(f)/(k''(f) log k(f))), is based on our improvement of a bound by Chattopadhyay, Hatami, Lovett and Tal (ITCS, 2019) on the sum of absolute values of level-1 Fourier coefficients in terms of 𝔽₂-degree. We further show that Chang’s lemma for the above-mentioned choices of the threshold is asymptotically outperformed by our bounds for most settings of the parameters involved.
Next, we show that our bounds are tight for a wide range of the parameters involved, by constructing functions witnessing their tightness. All the functions we construct are modifications of the Addressing function, where we replace certain input variables by suitable functions. Our final contribution is to construct Boolean functions f for which our lower bounds asymptotically match δ(f), and for any choice of the threshold t, the lower bound obtained from Chang’s lemma is asymptotically smaller than δ(f).
Our results imply more refined deterministic one-way communication complexity upper bounds for XOR functions. Given the wide-ranging application of Chang’s lemma to areas like additive combinatorics, learning theory and communication complexity, we strongly feel that our refinements of Chang’s lemma will find many more applications.

Sourav Chakraborty, Nikhil S. Mande, Rajat Mittal, Tulasimohan Molli, Manaswi Paraashar, and Swagato Sanyal. Tight Chang’s-Lemma-Type Bounds for Boolean Functions. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2021.10, author = {Chakraborty, Sourav and Mande, Nikhil S. and Mittal, Rajat and Molli, Tulasimohan and Paraashar, Manaswi and Sanyal, Swagato}, title = {{Tight Chang’s-Lemma-Type Bounds for Boolean Functions}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {10:1--10:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.10}, URN = {urn:nbn:de:0030-drops-155215}, doi = {10.4230/LIPIcs.FSTTCS.2021.10}, annote = {Keywords: Analysis of Boolean functions, Chang’s lemma, Parity decision trees, Fourier dimension} }

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APPROX

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

In this work, we resolve the query complexity of global minimum cut problem for a graph by designing a randomized algorithm for approximating the size of minimum cut in a graph, where the graph can be accessed through local queries like Degree, Neighbor, and Adjacency queries.
Given ε ∈ (0,1), the algorithm with high probability outputs an estimate t̂ satisfying the following (1-ε) t ≤ t̂ ≤ (1+ε) t, where t is the size of minimum cut in the graph. The expected number of local queries used by our algorithm is min{m+n,m/t}poly(log n,1/(ε)) where n and m are the number of vertices and edges in the graph, respectively. Eden and Rosenbaum showed that Ω(m/t) local queries are required for approximating the size of minimum cut in graphs, {but no local query based algorithm was known. Our algorithmic result coupled with the lower bound of Eden and Rosenbaum [APPROX 2018] resolve the query complexity of the problem of estimating the size of minimum cut in graphs using local queries.}
Building on the lower bound of Eden and Rosenbaum, we show that, for all t ∈ ℕ, Ω(m) local queries are required to decide if the size of the minimum cut in the graph is t or t-2. Also, we show that, for any t ∈ ℕ, Ω(m) local queries are required to find all the minimum cut edges even if it is promised that the input graph has a minimum cut of size t. Both of our lower bound results are randomized, and hold even if we can make Random Edge queries in addition to local queries.

Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar. Query Complexity of Global Minimum Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bishnu_et_al:LIPIcs.APPROX/RANDOM.2021.6, author = {Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath and Paraashar, Manaswi}, title = {{Query Complexity of Global Minimum Cut}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {6:1--6:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.6}, URN = {urn:nbn:de:0030-drops-146992}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.6}, annote = {Keywords: Query complexity, Global mincut} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

The disjointness problem - where Alice and Bob are given two subsets of {1, … , n} and they have to check if their sets intersect - is a central problem in the world of communication complexity. While both deterministic and randomized communication complexities for this problem are known to be Θ(n), it is also known that if the sets are assumed to be drawn from some restricted set systems then the communication complexity can be much lower. In this work, we explore how communication complexity measures change with respect to the complexity of the underlying set system. The complexity measure for the set system that we use in this work is the Vapnik–Chervonenkis (VC) dimension. More precisely, on any set system with VC dimension bounded by d, we analyze how large can the deterministic and randomized communication complexities be, as a function of d and n. The d-sparse set disjointness problem, where the sets have size at most d, is one such set system with VC dimension d. The deterministic and the randomized communication complexities of the d-sparse set disjointness problem have been well studied and is known to be Θ (d log ({n}/{d})) and Θ(d), respectively, in the multi-round communication setting. In this paper, we address the question of whether the randomized communication complexity is always upper bounded by a function of the VC dimension of the set system, and does there always exist a gap between the deterministic and randomized communication complexity for set systems with small VC dimension.
In this paper, we construct two natural set systems of VC dimension d, motivated from geometry. Using these set systems we show that the deterministic and randomized communication complexity can be Θ̃(dlog (n/d)) for set systems of VC dimension d and this matches the deterministic upper bound for all set systems of VC dimension d. We also study the deterministic and randomized communication complexities of the set intersection problem when sets belong to a set system of bounded VC dimension. We show that there exists set systems of VC dimension d such that both deterministic and randomized (one-way and multi-round) complexities for the set intersection problem can be as high as Θ(dlog (n/d)), and this is tight among all set systems of VC dimension d.

Anup Bhattacharya, Sourav Chakraborty, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar. Disjointness Through the Lens of Vapnik–Chervonenkis Dimension: Sparsity and Beyond. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhattacharya_et_al:LIPIcs.APPROX/RANDOM.2020.23, author = {Bhattacharya, Anup and Chakraborty, Sourav and Ghosh, Arijit and Mishra, Gopinath and Paraashar, Manaswi}, title = {{Disjointness Through the Lens of Vapnik–Chervonenkis Dimension: Sparsity and Beyond}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.23}, URN = {urn:nbn:de:0030-drops-126261}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.23}, annote = {Keywords: Communication complexity, VC dimension, Sparsity, and Geometric Set System} }

Document

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

Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function f:{-1,1}ⁿ → {-1,1} and •:{-1,1}² → {-1,1} the two-party bounded-error quantum communication complexity of (f ∘ •) is O(Q(f) log n), where Q(f) is the bounded-error quantum query complexity of f. Note that the bounded-error randomized communication complexity of (f ∘ •) is bounded by O(R(f)), where R(f) denotes the bounded-error randomized query complexity of f. Thus, the BCW simulation has an extra O(log n) factor appearing that is absent in classical simulation. A natural question is if this factor can be avoided. Razborov (IZV MATH'03) showed that the bounded-error quantum communication complexity of Set-Disjointness is Ω(√n). The BCW simulation yields an upper bound of O(√n log n). Høyer and de Wolf (STACS'02) showed that this can be reduced to c^(log^* n) for some constant c, and subsequently Aaronson and Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the quantum communication complexity of the Set-Disjointness function (which is NOR_n ∘ ∧) is O(Q(NOR_n)).
Perhaps somewhat surprisingly, we show that when • = ⊕, then the extra log n factor in the BCW simulation is unavoidable. In other words, we exhibit a total function F:{-1,1}ⁿ → {-1,1} such that Q^{cc}(F ∘ ⊕) = Θ(Q(F) log n).
To the best of our knowledge, it was not even known prior to this work whether there existed a total function F and 2-bit function •, such that Q^{cc}(F ∘ •) = ω(Q(F)).

Sourav Chakraborty, Arkadev Chattopadhyay, Nikhil S. Mande, and Manaswi Paraashar. Quantum Query-To-Communication Simulation Needs a Logarithmic Overhead. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 32:1-32:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chakraborty_et_al:LIPIcs.CCC.2020.32, author = {Chakraborty, Sourav and Chattopadhyay, Arkadev and Mande, Nikhil S. and Paraashar, Manaswi}, title = {{Quantum Query-To-Communication Simulation Needs a Logarithmic Overhead}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {32:1--32:15}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.32}, URN = {urn:nbn:de:0030-drops-125842}, doi = {10.4230/LIPIcs.CCC.2020.32}, annote = {Keywords: Quantum query complexity, quantum communication complexity, approximate degree, approximate spectral norm} }

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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)

We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k-Fourier-sparse n-bit Boolean function from O(k^{1.5}(log k)^2) uniform quantum examples for that function. This improves over the bound of Theta~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang’s lemma for sparse Boolean functions. Second, we show that if a concept class {C} can be exactly learned using Q quantum membership queries, then it can also be learned using O ({Q^2}/{log Q} * log|C|) classical membership queries. This improves the previous-best simulation result (Servedio-Gortler, SICOMP'04) by a log Q-factor.

Srinivasan Arunachalam, Sourav Chakraborty, Troy Lee, Manaswi Paraashar, and Ronald de Wolf. Two New Results About Quantum Exact Learning. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{arunachalam_et_al:LIPIcs.ICALP.2019.16, author = {Arunachalam, Srinivasan and Chakraborty, Sourav and Lee, Troy and Paraashar, Manaswi and de Wolf, Ronald}, title = {{Two New Results About Quantum Exact Learning}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {16:1--16:15}, 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.2019.16}, URN = {urn:nbn:de:0030-drops-105929}, doi = {10.4230/LIPIcs.ICALP.2019.16}, annote = {Keywords: quantum computing, exact learning, analysis of Boolean functions, Fourier sparse Boolean functions} }

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