Document

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

Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities is an outstanding problem that has received a lot of attention after its introduction by Itsykson and Sokolov [Dmitry Itsykson and Dmitry Sokolov, 2014]. Very recently, Efremenko, Garlík and Itsykson [Klim Efremenko et al., 2023] proved the first exponential lower bounds on the size of ResLin proofs that were additionally restricted to be bottom-regular. We show that there are formulas for which such regular ResLin proofs of unsatisfiability continue to have exponential size even though there exist short proofs of their unsatisfiability in ordinary, non-regular resolution. This is the first super-polynomial separation between the power of general ResLin and that of regular ResLin for any natural notion of regularity.
Our argument, while building upon the work of Efremenko et al. [Klim Efremenko et al., 2023], uses additional ideas from the literature on lifting theorems.

Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák. Exponential Separation Between Powers of Regular and General Resolution over Parities. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 23:1-23:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bhattacharya_et_al:LIPIcs.CCC.2024.23, author = {Bhattacharya, Sreejata Kishor and Chattopadhyay, Arkadev and Dvo\v{r}\'{a}k, Pavel}, title = {{Exponential Separation Between Powers of Regular and General Resolution over Parities}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {23:1--23:32}, 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.2024.23}, URN = {urn:nbn:de:0030-drops-204191}, doi = {10.4230/LIPIcs.CCC.2024.23}, annote = {Keywords: Proof Complexity, Regular Reslin, Branching Programs, Lifting} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

We relate various complexity measures like sensitivity, block sensitivity, certificate complexity for multi-output functions to the query complexities of such functions. Using these relations, we provide the following improvements upon the known relationship between pseudo-deterministic and deterministic query complexity for total search problems:
- We show that deterministic query complexity is at most the third power of its pseudo-deterministic query complexity. Previously, a fourth-power relation was shown by Goldreich, Goldwasser and Ron (ITCS'13).
- We improve the known separation between pseudo-deterministic and randomized decision tree size for total search problems in two ways: (1) we exhibit an exp(Ω̃(n^{1/4})) separation for the SearchCNF relation for random k-CNFs. This seems to be the first exponential lower bound on the pseudo-deterministic size complexity of SearchCNF associated with random k-CNFs. (2) we exhibit an exp(Ω(n)) separation for the ApproxHamWt relation. The previous best known separation for any relation was exp(Ω(n^{1/2})). We also separate pseudo-determinism from randomness in And and (And,Or) decision trees, and determinism from pseudo-determinism in Parity decision trees. For a hypercube colouring problem, that was introduced by Goldwasswer, Impagliazzo, Pitassi and Santhanam (CCC'21) to analyze the pseudo-deterministic complexity of a complete problem in TFNP^{dt}, we prove that either the monotone block-sensitivity or the anti-monotone block sensitivity is Ω(n^{1/3}); Goldwasser et al. showed an Ω(n^{1/2}) bound for general block-sensitivity.

Arkadev Chattopadhyay, Yogesh Dahiya, and Meena Mahajan. Query Complexity of Search Problems. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chattopadhyay_et_al:LIPIcs.MFCS.2023.34, author = {Chattopadhyay, Arkadev and Dahiya, Yogesh and Mahajan, Meena}, title = {{Query Complexity of Search Problems}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.34}, URN = {urn:nbn:de:0030-drops-185689}, doi = {10.4230/LIPIcs.MFCS.2023.34}, annote = {Keywords: Decision trees, Search problems, Pseudo-determinism, Randomness} }

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

We show that the deterministic decision tree complexity of a (partial) function or relation f lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation f∘g as long as the gadget g satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([Göös, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of f, which could be exponentially smaller than its deterministic counterpart when either f is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to f. Reducing the gadget size to a constant is an important open problem at the frontier of current research.
Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., Res(⊕), of the unsatisfiability of closely related constant-width CNF formulas.

Arkadev Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif. Lifting to Parity Decision Trees via Stifling. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2023.33, author = {Chattopadhyay, Arkadev and Mande, Nikhil S. and Sanyal, Swagato and Sherif, Suhail}, title = {{Lifting to Parity Decision Trees via Stifling}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {33:1--33:20}, 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.2023.33}, URN = {urn:nbn:de:0030-drops-175362}, doi = {10.4230/LIPIcs.ITCS.2023.33}, annote = {Keywords: Decision trees, parity decision trees, lifting theorems} }

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

We establish an ε-sensitive hierarchy separation for monotone arithmetic computations. The notion of ε-sensitive monotone lower bounds was recently introduced by Hrubeš [Pavel Hrubeš, 2020]. We show the following:
- There exists a monotone polynomial over n variables in VNP that cannot be computed by 2^o(n) size monotone circuits in an ε-sensitive way as long as ε ≥ 2^(-Ω(n)).
- There exists a polynomial over n variables that can be computed by polynomial size monotone circuits but cannot be computed by any monotone arithmetic branching program (ABP) of n^o(log n) size, even in an ε-sensitive fashion as long as ε ≥ n^(-Ω(log n)).
- There exists a polynomial over n variables that can be computed by polynomial size monotone ABPs but cannot be computed in n^o(log n) size by monotone formulas even in an ε-sensitive way, when ε ≥ n^(-Ω(log n)).
- There exists a polynomial over n variables that can be computed by width-4 polynomial size monotone arithmetic branching programs (ABPs) but cannot be computed in 2^o(n^{1/d}) size by monotone, unbounded fan-in formulas of product depth d even in an ε-sensitive way, when ε ≥ 2^(-Ω(n^{1/d})). This yields an ε-sensitive separation of constant-depth monotone formulas and constant-width monotone ABPs. The novel feature of our separations is that in each case the polynomial exhibited is obtained from a graph inner-product polynomial by choosing an appropriate graph topology. The closely related graph inner-product Boolean function for expander graphs was invented by Hayes [Thomas P. Hayes, 2011], also independently by Pitassi [Toniann Pitassi, 2009], in the context of best-partition multiparty communication complexity.

Arkadev Chattopadhyay, Utsab Ghosal, and Partha Mukhopadhyay. Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product. 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. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2022.12, author = {Chattopadhyay, Arkadev and Ghosal, Utsab and Mukhopadhyay, Partha}, title = {{Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {12:1--12:20}, 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.2022.12}, URN = {urn:nbn:de:0030-drops-174045}, doi = {10.4230/LIPIcs.FSTTCS.2022.12}, annote = {Keywords: Algebraic Complexity, Discrepancy, Lower Bounds, Monotone Computations} }

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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 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

We prove two results that shed new light on the monotone complexity of the spanning tree polynomial, a classic polynomial in algebraic complexity and beyond.
First, we show that the spanning tree polynomials having n variables and defined over constant-degree expander graphs, have monotone arithmetic complexity 2^{Ω(n)}. This yields the first strongly exponential lower bound on monotone arithmetic circuit complexity for a polynomial in VP. Before this result, strongly exponential size monotone lower bounds were known only for explicit polynomials in VNP [S. B. Gashkov and I. S. Sergeev, 2012; Ran Raz and Amir Yehudayoff, 2011; Srikanth Srinivasan, 2020; Bruno Pasqualotto Cavalar et al., 2020; Pavel Hrubeš and Amir Yehudayoff, 2021].
Recently, Hrubeš [Pavel Hrubeš, 2020] initiated a program to prove lower bounds against general arithmetic circuits by proving ε-sensitive lower bounds for monotone arithmetic circuits for a specific range of values for ε ∈ (0,1). The first ε-sensitive lower bound was just proved for a family of polynomials inside VNP by Chattopadhyay, Datta and Mukhopadhyay [Arkadev Chattopadhyay et al., 2021]. We consider the spanning tree polynomial ST_n defined over the complete graph of n vertices and show that the polynomials F_{n-1,n} - ε⋅ ST_{n} and F_{n-1,n} + ε⋅ ST_{n}, defined over (n-1)n variables, have monotone circuit complexity 2^{Ω(n)} if ε ≥ 2^{- Ω(n)} and F_{n-1,n} := ∏_{i = 2}ⁿ (x_{i,1} + ⋯ + x_{i,n}) is the complete set-multilinear polynomial. This provides the first ε-sensitive exponential lower bound for a family of polynomials inside VP. En-route, we consider a problem in 2-party, best partition communication complexity of deciding whether two sets of oriented edges distributed among Alice and Bob form a spanning tree or not. We prove that there exists a fixed distribution, under which the problem has low discrepancy with respect to every nearly-balanced partition. This result could be of interest beyond algebraic complexity.
Our two results, thus, are incomparable generalizations of the well known result by Jerrum and Snir [Mark Jerrum and Marc Snir, 1982] which showed that the spanning tree polynomial, defined over complete graphs with n vertices (so the number of variables is (n-1)n), has monotone complexity 2^{Ω(n)}. In particular, the first result is an optimal lower bound and the second result can be thought of as a robust version of the earlier monotone lower bound for the spanning tree polynomial.

Arkadev Chattopadhyay, Rajit Datta, Utsab Ghosal, and Partha Mukhopadhyay. Monotone Complexity of Spanning Tree Polynomial Re-Visited. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 39:1-39:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ITCS.2022.39, author = {Chattopadhyay, Arkadev and Datta, Rajit and Ghosal, Utsab and Mukhopadhyay, Partha}, title = {{Monotone Complexity of Spanning Tree Polynomial Re-Visited}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {39:1--39:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.39}, URN = {urn:nbn:de:0030-drops-156356}, doi = {10.4230/LIPIcs.ITCS.2022.39}, annote = {Keywords: Spanning Tree Polynomial, Monotone Computation, Lower Bounds, Communication Complexity} }

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

We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n³) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [Chattopadhyay et al., 2020] both qualitatively (in terms of designing a large number of examples) and quantitatively (shrinking the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).

Arkadev Chattopadhyay, Ankit Garg, and Suhail Sherif. Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture. 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. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2021.13, author = {Chattopadhyay, Arkadev and Garg, Ankit and Sherif, Suhail}, title = {{Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {13:1--13:16}, 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.2021.13}, URN = {urn:nbn:de:0030-drops-155245}, doi = {10.4230/LIPIcs.FSTTCS.2021.13}, annote = {Keywords: Approximate Rank, Randomized Parity Decision Trees, Randomized Communication Complexity, XOR functions, Subspace Designs} }

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**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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Complete Volume

**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

LIPIcs, Volume 150, FSTTCS'19, Complete Volume

39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2019, title = {{LIPIcs, Volume 150, FSTTCS'19, Complete Volume}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019}, URN = {urn:nbn:de:0030-drops-116426}, doi = {10.4230/LIPIcs.FSTTCS.2019}, annote = {Keywords: Theory of computation} }

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Front Matter

**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Front Matter, Table of Contents, Preface, Conference Organization

39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2019.0, author = {Chattopadhyay, Arkadev and Gastin, Paul}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {0:i--0:xii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.0}, URN = {urn:nbn:de:0030-drops-115621}, doi = {10.4230/LIPIcs.FSTTCS.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

The canonical problem that gives an exponential separation between deterministic and randomized communication complexity in the classical two-party communication model is "Equality". In this work we show that even allowing access to an "Equality" oracle, deterministic protocols remain exponentially weaker than randomized ones. More precisely, we exhibit a total function on n bits with randomized one-sided communication complexity O(log n), but such that every deterministic protocol with access to "Equality" oracle needs Omega(n) cost to compute it.
Additionally we exhibit a natural and strict infinite hierarchy within BPP, starting with the class P^{EQ} at its bottom.

Arkadev Chattopadhyay, Shachar Lovett, and Marc Vinyals. Equality Alone Does not Simulate Randomness. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 14:1-14:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2019.14, author = {Chattopadhyay, Arkadev and Lovett, Shachar and Vinyals, Marc}, title = {{Equality Alone Does not Simulate Randomness}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {14:1--14:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.14}, URN = {urn:nbn:de:0030-drops-108368}, doi = {10.4230/LIPIcs.CCC.2019.14}, annote = {Keywords: Communication lower bound, derandomization} }

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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 prove a new query-to-communication lifting for randomized protocols, with inner product as gadget. This allows us to use a much smaller gadget, leading to a more efficient lifting. Prior to this work, such a theorem was known only for deterministic protocols, due to Chattopadhyay et al. [Arkadev Chattopadhyay et al., 2017] and Wu et al. [Xiaodi Wu et al., 2017]. The only query-to-communication lifting result for randomized protocols, due to Göös, Pitassi and Watson [Mika Göös et al., 2017], used the much larger indexing gadget.
Our proof also provides a unified treatment of randomized and deterministic lifting. Most existing proofs of deterministic lifting theorems use a measure of information known as thickness. In contrast, Göös, Pitassi and Watson [Mika Göös et al., 2017] used blockwise min-entropy as a measure of information. Our proof uses the blockwise min-entropy framework to prove lifting theorems in both settings in a unified way.

Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, and Toniann Pitassi. Query-To-Communication Lifting for BPP Using Inner Product. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chattopadhyay_et_al:LIPIcs.ICALP.2019.35, author = {Chattopadhyay, Arkadev and Filmus, Yuval and Koroth, Sajin and Meir, Or and Pitassi, Toniann}, title = {{Query-To-Communication Lifting for BPP Using Inner Product}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {35:1--35:15}, 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.2019.35}, URN = {urn:nbn:de:0030-drops-106110}, doi = {10.4230/LIPIcs.ICALP.2019.35}, annote = {Keywords: lifting theorems, inner product, BPP Lifting, Deterministic Lifting} }

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**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

We use a technique of “lifting” functions introduced by Krause and Pudlak [Theor. Comput. Sci., 1997], to amplify
degree-hardness measures of a function to corresponding monomial-hardness properties of the
lifted function. We then show that any symmetric function F projects onto a “lift” of another
suitable symmetric function f . These two key results enable us to prove several results on the
complexity of symmetric functions in various models, as given below:
1. We provide a characterization of the approximate spectral norm of symmetric functions in
terms of the spectrum of the underlying predicate, affirming a conjecture of Ada et al. [APPROX-RANDOM, 2012]
which has several consequences.
2. We characterize symmetric functions computable by quasi-polynomial sized Threshold
of Parity circuits.
3. We show that the approximate spectral norm of a symmetric function f characterizes the
(quantum and classical) bounded error communication complexity of f o XOR.
4. Finally, we characterize the weakly-unbounded error communication complexity of symmetric
XOR functions, resolving a weak form of a conjecture by Shi and Zhang [Quantum Information & Computation, 2009]

Arkadev Chattopadhyay and Nikhil S. Mande. A Lifting Theorem with Applications to Symmetric Functions. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2017.23, author = {Chattopadhyay, Arkadev and Mande, Nikhil S.}, title = {{A Lifting Theorem with Applications to Symmetric Functions}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {23:1--23:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.23}, URN = {urn:nbn:de:0030-drops-83839}, doi = {10.4230/LIPIcs.FSTTCS.2017.23}, annote = {Keywords: Symmetric functions, lifting, circuit complexity, communication com- plexity} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let f: {0,1}^2n -> {0,1} be any boolean function. Alice and Bob get k inputs x_1, ..., x_k and y_1, ..., y_k respectively, with x_i,y_i in {0,1}^n. They want to output a k-bit vector v, such that there exists one index i for which v_i is not equal f(x_i,y_i). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds.
To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola, to show that Greater-Than is weakly regular.

Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Lower Bounds for Elimination via Weak Regularity. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chattopadhyay_et_al:LIPIcs.STACS.2017.21, author = {Chattopadhyay, Arkadev and Dvor\'{a}k, Pavel and Kouck\'{y}, Michal and Loff, Bruno and Mukhopadhyay, Sagnik}, title = {{Lower Bounds for Elimination via Weak Regularity}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.21}, URN = {urn:nbn:de:0030-drops-70128}, doi = {10.4230/LIPIcs.STACS.2017.21}, annote = {Keywords: communication complexity, elimination, discrepancy, regularity, greater-than} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We consider the point-to-point message passing model of communication in which there are $k$ processors with individual private inputs, each n-bit long. Each processor is located at the node of an underlying undirected graph and has access to private random coins. An edge of the graph is a private channel of communication between its endpoints. The processors have to compute a given function of all their inputs by communicating along these channels. While this model has been widely used in distributed computing, strong lower bounds on the amount of communication needed to compute simple functions have just begun to appear.
In this work, we prove a tight lower bound of \Omega(kn) on the communication needed for computing the Tribes function, when the underlying graph is a star of k+1 nodes that has k leaves with inputs and a center with no input. A lower bound on this topology easily implies comparable bounds for others. Our lower bounds are obtained by building upon the recent information theoretic techniques of Braverman et al. ([4], FOCS'13) and combining it with the earlier work of Jayram, Kumar and Sivakumar ([10], STOC'03). This approach yields information complexity bounds that are of independent interest.

Arkadev Chattopadhyay and Sagnik Mukhopadhyay. Tribes Is Hard in the Message Passing Model. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 224-237, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{chattopadhyay_et_al:LIPIcs.STACS.2015.224, author = {Chattopadhyay, Arkadev and Mukhopadhyay, Sagnik}, title = {{Tribes Is Hard in the Message Passing Model}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {224--237}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.224}, URN = {urn:nbn:de:0030-drops-49162}, doi = {10.4230/LIPIcs.STACS.2015.224}, annote = {Keywords: communication complexity, Tribes, information complexity, direct-sum} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

In the `Number-on-Forehead' (NOF) model of multiparty communication,
the input is a k times m boolean matrix A (where k is the number of players) and Player i sees all bits except those in the i-th row, and the players communicate by broadcast in order to evaluate a specified function f at A.
We discover new computational power when k exceeds log m. We give a protocol with communication cost poly-logarithmic in m, for block composed functions with limited block width. These are functions
of the form f o g where f is a symmetric b-variate function, and g is a (kr)-variate function and (f o g)(A) is defined, for a k times (br) matrix to be f(g(A-1),...,g(A-b)) where A-i is the i-th (k times r) block of A. Our protocol works provided that k > 1+ ln b + (2 to the power of r).
Ada et al. (ICALP'2012) previously obtained simultaneous and deterministic efficient protocols for composed functions of block-width one. The new protocol is the first to work for block composed functions with block-width greather than one. Moreover, it is simultaneous, with vanishingly small error probability, if public coin randomness is allowed. The deterministic and zero-error version barely uses interaction.

Arkadev Chattopadhyay and Michael E. Saks. The Power of Super-logarithmic Number of Players. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 596-603, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{chattopadhyay_et_al:LIPIcs.APPROX-RANDOM.2014.596, author = {Chattopadhyay, Arkadev and Saks, Michael E.}, title = {{The Power of Super-logarithmic Number of Players}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {596--603}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.596}, URN = {urn:nbn:de:0030-drops-47243}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.596}, annote = {Keywords: Communication complexity, Number-On-Forehead model, composed functions} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with $O(\log\log n)$ depth and $O(\log^2 n)$ nondeterministic bits,
where $n$ is the number of group elements. This improves the existing upper bound from \cite{Wolf 94} for the problems. In the previous upper bound the circuits have bounded fan-in but depth $O(\log^2 n)$ and also $O(\log^2 n)$ nondeterministic bits. We then prove that the kind of circuits from our upper bound cannot compute the Parity function. Since Parity is AC0 reducible to Graph Isomorphism, this implies that Graph Isomorphism is strictly harder than Group or Quasigroup Isomorphism under the ordering defined by AC0 reductions.

Arkadev Chattopadhyay, Jacobo Torán, and Fabian Wagner. Graph Isomorphism is not AC^0 reducible to Group Isomorphism. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 317-326, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2010.317, author = {Chattopadhyay, Arkadev and Tor\'{a}n, Jacobo and Wagner, Fabian}, title = {{Graph Isomorphism is not AC^0 reducible to Group Isomorphism}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {317--326}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.317}, URN = {urn:nbn:de:0030-drops-28748}, doi = {10.4230/LIPIcs.FSTTCS.2010.317}, annote = {Keywords: Complexity, Algorithms, Group Isomorphism Problem, Circuit Com plexity} }