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

The identity testing of rational formulas (RIT) in the free skew field efficiently reduces to computing the rank of a matrix whose entries are linear polynomials in noncommuting variables [Hrubeš and Wigderson, 2015]. This rank computation problem has deterministic polynomial-time white-box algorithms [Ankit Garg et al., 2016; Ivanyos et al., 2018] and a randomized polynomial-time algorithm in the black-box setting [Harm Derksen and Visu Makam, 2017]. In this paper, we propose a new approach for efficient derandomization of black-box RIT. Additionally, we obtain results for matrix rank computation over the free skew field and construct efficient linear pencil representations for a new class of rational expressions. More precisely, we show:
- Under the hardness assumption that the ABP (algebraic branching program) complexity of every polynomial identity for the k×k matrix algebra is 2^Ω(k) [Andrej Bogdanov and Hoeteck Wee, 2005], we obtain a subexponential-time black-box RIT algorithm for rational formulas of inversion height almost logarithmic in the size of the formula. This can be seen as the first "hardness implies derandomization" type theorem for rational formulas.
- We show that the noncommutative rank of any matrix over the free skew field whose entries have small linear pencil representations can be computed in deterministic polynomial time. While an efficient rank computation was known for matrices with noncommutative formulas as entries [Ankit Garg et al., 2020], we obtain the first deterministic polynomial-time algorithms for rank computation of matrices whose entries are noncommutative ABPs or rational formulas.
- Motivated by the definition given by Bergman [George M Bergman, 1976], we define a new class of rational functions where a rational function of inversion height at most h is defined as a composition of a noncommutative r-skewed circuit (equivalently an ABP) with inverses of rational functions of this class of inversion height at most h-1 which are also disjoint. We obtain a polynomial-size linear pencil representation for this class which gives a white-box deterministic polynomial-time identity testing algorithm for the class.

V. Arvind, Abhranil Chatterjee, Utsab Ghosal, Partha Mukhopadhyay, and C. Ramya. On Identity Testing and Noncommutative Rank Computation over the Free Skew Field. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arvind_et_al:LIPIcs.ITCS.2023.6, author = {Arvind, V. and Chatterjee, Abhranil and Ghosal, Utsab and Mukhopadhyay, Partha and Ramya, C.}, title = {{On Identity Testing and Noncommutative Rank Computation over the Free Skew Field}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {6:1--6:23}, 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.2023.6}, URN = {urn:nbn:de:0030-drops-175093}, doi = {10.4230/LIPIcs.ITCS.2023.6}, annote = {Keywords: Algebraic Complexity, Identity Testing, Non-commutative rank} }

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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 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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