21 Search Results for "Mukhopadhyay, Partha"

Document
DOI: 10.4230/LIPIcs.ITCS.2026.49
Debordering Closure Results in Determinantal and Pfaffian Ideals

Authors: Anakin Dey and Zeyu Guo

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
One important question in algebraic complexity is understanding the complexity of polynomial ideals (Grochow, Bulletin of EATCS 131, 2020). Andrews and Forbes (STOC 2022) studied the determinantal ideals I^{det}_{n,m,r} generated by the r× r minors of n× m matrices. Over fields of characteristic zero or of sufficiently large characteristic, they showed that for any nonzero f ∈ I^{det}_{n,m,r}, the determinant of a t × t matrix of variables with t = Θ{r^{1/3}} is approximately computed by a constant-depth, polynomial-size f-oracle algebraic circuit, in the sense that the determinant lies in the border of such circuits. An analogous result was also obtained for Pfaffians in the same paper. In this work, we deborder the result of Andrews and Forbes by showing that when f has polynomial degree, the determinant is in fact exactly computed by a constant-depth, polynomial-size f-oracle algebraic circuit. We further establish an analogous result for Pfaffian ideals. Our results are established using the isolation lemma, combined with a careful analysis of straightening-law expansions of polynomials in determinantal and Pfaffian ideals.
Cite as

Anakin Dey and Zeyu Guo. Debordering Closure Results in Determinantal and Pfaffian Ideals. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 49:1-49:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dey_et_al:LIPIcs.ITCS.2026.49,
  author =	{Dey, Anakin and Guo, Zeyu},
  title =	{{Debordering Closure Results in Determinantal and Pfaffian Ideals}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{49:1--49:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.49},
  URN =		{urn:nbn:de:0030-drops-253363},
  doi =		{10.4230/LIPIcs.ITCS.2026.49},
  annote =	{Keywords: Algebraic circuit complexity, Isolation lemma, Debordering}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.3
Linear Matroid Intersection Is in Catalytic Logspace

Authors: Aryan Agarwala, Yaroslav Alekseev, and Antoine Vinciguerra

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Linear matroid intersection is an important problem in combinatorial optimization. Given two linear matroids over the same ground set, the linear matroid intersection problem asks you to find a common independent set of maximum size. The deep interest in linear matroid intersection is due to the fact that it generalises many classical problems in theoretical computer science, such as bipartite matching, edge disjoint spanning trees, rainbow spanning tree, and many more. We study this problem in the model of catalytic computation: space-bounded machines are granted access to catalytic space, which is additional working memory that is full with arbitrary data that must be preserved at the end of its computation. Although linear matroid intersection has had a polynomial time algorithm for over 50 years, it remains an important open problem to show that linear matroid intersection belongs to any well studied subclass of {P}. We address this problem for the class catalytic logspace (CL) with a polynomial time bound (CLP). Recently, Agarwala and Mertz (2025) showed that bipartite maximum matching can be computed in the class CLP ⊆ {P}. This was the first subclass of {P} shown to contain bipartite matching, and additionally the first problem outside TC¹ shown to be contained in CL. We significantly improve the result of Agarwala and Mertz by showing that linear matroid intersection can be computed in CLP.
Cite as

Aryan Agarwala, Yaroslav Alekseev, and Antoine Vinciguerra. Linear Matroid Intersection Is in Catalytic Logspace. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 3:1-3:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{agarwala_et_al:LIPIcs.ITCS.2026.3,
  author =	{Agarwala, Aryan and Alekseev, Yaroslav and Vinciguerra, Antoine},
  title =	{{Linear Matroid Intersection Is in Catalytic Logspace}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{3:1--3:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.3},
  URN =		{urn:nbn:de:0030-drops-252908},
  doi =		{10.4230/LIPIcs.ITCS.2026.3},
  annote =	{Keywords: Catalytic Computing, Computational Complexity, Matroid Theory, Algorithms}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2025.51
Randomized Black-Box PIT for Small Depth +-Regular Non-Commutative Circuits

Authors: G. V. Sumukha Bharadwaj and S. Raja

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract
In this paper, we address the black-box polynomial identity testing (PIT) problem for non-commutative polynomials computed by +-regular circuits, a class of homogeneous circuits introduced by Arvind, Joglekar, Mukhopadhyay, and Raja (STOC 2017, Theory of Computing 2019). These circuits can compute polynomials with a number of monomials that are doubly exponential in the circuit size. They gave an efficient randomized PIT algorithm for +-regular circuits of depth 3 and posed the problem of developing an efficient black-box PIT for higher depths as an open problem. Our work makes progress on this open problem by resolving it for constant-depth +-regular circuits. We present a randomized black-box polynomial-time algorithm for +-regular circuits of any constant depth. Specifically, our algorithm runs in s^{O(d²)} time, where s and d represent the size and the depth of the +-regular circuit, respectively. Our approach combines several key techniques in a novel way. We employ a nondeterministic substitution automaton that transforms the polynomial into a structured form and utilizes polynomial sparsification along with commutative transformations to maintain non-zeroness. Additionally, we introduce matrix composition, coefficient modification via the automaton, and multi-entry outputs - methods that have not previously been applied in the context of black-box PIT. Together, these techniques enable us to effectively handle exponential degrees and doubly exponential sparsity in non-commutative settings, enabling polynomial identity testing for higher-depth circuits. In particular, we show that if f is a non-zero non-commutative polynomial in n variables over the field 𝔽, computed by a depth-d +-regular circuit of size s, then f cannot be a polynomial identity for the matrix algebra 𝕄_{N}(𝔽), where N = s^{O(d²)} and the size of the field 𝔽 depends on the degree of f. Interestingly, the size of the matrices does not depend on the degree of f. Our result can be interpreted as an Amitsur-Levitzki-type result [Amitsur and Levitzki, 1950] for polynomials computed by small-depth +-regular circuits.
Cite as

G. V. Sumukha Bharadwaj and S. Raja. Randomized Black-Box PIT for Small Depth +-Regular Non-Commutative Circuits. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{sumukhabharadwaj_et_al:LIPIcs.FSTTCS.2025.51,
  author =	{Sumukha Bharadwaj, G. V. and Raja, S.},
  title =	{{Randomized Black-Box PIT for Small Depth +-Regular Non-Commutative Circuits}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{51:1--51:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.51},
  URN =		{urn:nbn:de:0030-drops-250949},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.51},
  annote =	{Keywords: Polynomial Identity Testing, Non-commutative Circuits, Algebraic Circuits, +-Regular Circuits, Black-Box}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2025.22
IPS Lower Bounds for Formulas and Sum of ROABPs

Authors: Prerona Chatterjee, Utsab Ghosal, Partha Mukhopadhyay, and Amit Sinhababu

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract
We give new lower bounds for the fragments of the Ideal Proof System (IPS) introduced by Grochow and Pitassi [Joshua A. Grochow and Toniann Pitassi, 2018]. The Ideal Proof System is a central topic in algebraic proof complexity developed in the context of Nullstellensatz refutation [Paul Beame et al., 1994] and simulates Extended Frege efficiently. Our main results are as follows. - mult-IPS_{Lin'}: We prove nearly quadratic-size formula lower bound for multilinear refutation (over the Boolean hypercube) of a variant of the subset-sum axiom polynomial. Extending this, we obtain a nearly matching qualitative statement for a constant degree target polynomial. - IPS_{Lin'}: Over the fields of characteristic zero, we prove exponential-size sum-of-ROABPs lower bound for the refutation of a variant of the subset-sum axiom polynomial. The result also extends over the fields of positive characteristics when the target polynomial is suitably modified. The modification is inspired by the recent results [Tuomas Hakoniemi et al., 2024; Amik Raj Behera et al., 2025]. The mult-IPS_{Lin'} lower bound result is obtained by combining the quadratic-size formula lower bound technique of Kalorkoti [Kalorkoti, 1985] with some additional ideas. The proof technique of IPS_{Lin'} lower bound result is inspired by the recent lower bound result of Chatterjee, Kush, Saraf and Shpilka [Prerona Chatterjee et al., 2024].
Cite as

Prerona Chatterjee, Utsab Ghosal, Partha Mukhopadhyay, and Amit Sinhababu. IPS Lower Bounds for Formulas and Sum of ROABPs. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 22:1-22:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2025.22,
  author =	{Chatterjee, Prerona and Ghosal, Utsab and Mukhopadhyay, Partha and Sinhababu, Amit},
  title =	{{IPS Lower Bounds for Formulas and Sum of ROABPs}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{22:1--22:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.22},
  URN =		{urn:nbn:de:0030-drops-251035},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.22},
  annote =	{Keywords: Ideal Proof System, Lower Bound, Algebraic Complexity}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.56
Efficient Polynomial Identity Testing over Nonassociative Algebras

Authors: Partha Mukhopadhyay, C. Ramya, and Pratik Shastri

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We design the first efficient polynomial identity testing algorithms over the nonassociative polynomial algebra. In particular, multiplication among the formal variables is commutative but it is not associative. This complements the strong lower bound results obtained over this algebra by Hrubeš, Yehudayoff, and Wigderson [Pavel Hrubes et al., 2010] and Fijalkow, Lagarde, Ohlmann, and Serre [Fijalkow et al., 2021] from the identity testing perspective. Our main results are the following: - We construct nonassociative algebras (both commutative and noncommutative) which have no low degree identities. As a result, we obtain the first Amitsur-Levitzki type theorems [A. S. Amitsur and J. Levitzki, 1950] over nonassociative polynomial algebras. As a direct consequence, we obtain randomized polynomial-time black-box PIT algorithms for nonassociative polynomials which allow evaluation over such algebras. - On the derandomization side, we give a deterministic polynomial-time identity testing algorithm for nonassociative polynomials given by arithmetic circuits in the white-box setting. Previously, such an algorithm was known with the additional restriction of noncommutativity [Vikraman Arvind et al., 2017]. - In the black-box setting, we construct a hitting set of quasipolynomial-size for nonassociative polynomials computed by arithmetic circuits of small depth. Understanding the black-box complexity of identity testing, even in the randomized setting, was open prior to our work.
Cite as

Partha Mukhopadhyay, C. Ramya, and Pratik Shastri. Efficient Polynomial Identity Testing over Nonassociative Algebras. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 56:1-56:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mukhopadhyay_et_al:LIPIcs.APPROX/RANDOM.2025.56,
  author =	{Mukhopadhyay, Partha and C. Ramya and Shastri, Pratik},
  title =	{{Efficient Polynomial Identity Testing over Nonassociative Algebras}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{56:1--56:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.56},
  URN =		{urn:nbn:de:0030-drops-244224},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.56},
  annote =	{Keywords: Polynomial identity testing, nonassociative algebra, arithmetic circuits, black-box algorithms, white-box algorithms}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.9
Pseudorandom Bits for Non-Commutative Programs

Authors: Chin Ho Lee and Emanuele Viola

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
We obtain new explicit pseudorandom generators for several computational models involving groups. Our main results are as follows: 1) We consider read-once group-products over a finite group G, i.e., tests of the form ∏_{i=1}^n (g_i)^{x_i} where g_i ∈ G, a special case of read-once permutation branching programs. We give generators with optimal seed length c_G log(n/ε) over any p-group. The proof uses the small-bias plus noise paradigm, but derandomizes the noise to avoid the recursion in previous work. Our generator works when the bits are read in any order. Previously for any non-commutative group the best seed length was ≥ log n log(1/ε), even for a fixed order. 2) We give a reduction that "lifts" suitable generators for group products over G to a generator that fools width-w block products, i.e., tests of the form ∏ (g_i)^{f_i} where the f_i are arbitrary functions on disjoint blocks of w bits. Block products generalize several previously studied classes. The reduction applies to groups that are mixing in a representation-theoretic sense that we identify. 3) Combining (2) with (1) and other works we obtain new generators for block products over the quaternions or over any commutative group, with nearly optimal seed length. In particular, we obtain generators for read-once polynomials modulo any fixed m with nearly optimal seed length. Previously this was known only for m = 2. 4) We give a new generator for products over "mixing groups." The construction departs from previous work and uses representation theory. For constant error, we obtain optimal seed length, improving on previous work (which applied to any group). This paper identifies a challenge in the area that is reminiscent of a roadblock in circuit complexity - handling composite moduli - and points to several classes of groups to be attacked next.
Cite as

Chin Ho Lee and Emanuele Viola. Pseudorandom Bits for Non-Commutative Programs. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lee_et_al:LIPIcs.CCC.2025.9,
  author =	{Lee, Chin Ho and Viola, Emanuele},
  title =	{{Pseudorandom Bits for Non-Commutative Programs}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{9:1--9:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.9},
  URN =		{urn:nbn:de:0030-drops-237039},
  doi =		{10.4230/LIPIcs.CCC.2025.9},
  annote =	{Keywords: Group programs, Space-bounded derandomization, Representation theory}
}
Document
Research
DOI: 10.4230/TGDK.3.1.2
Talking Wikidata: Communication Patterns and Their Impact on Community Engagement in Collaborative Knowledge Graphs

Authors: Elisavet Koutsiana, Ioannis Reklos, Kholoud Saad Alghamdi, Nitisha Jain, Albert Meroño-Peñuela, and Elena Simperl

Published in: TGDK, Volume 3, Issue 1 (2025). Transactions on Graph Data and Knowledge, Volume 3, Issue 1

Abstract
We study collaboration patterns of Wikidata, one of the world's largest open source collaborative knowledge graph (KG) communities. Collaborative KG communities, play a key role in structuring machine-readable knowledge to support AI systems like conversational agents. However, these communities face challenges related to long-term member engagement, as a small subset of contributors often is responsible for the majority of contributions and decision-making. While prior research has explored contributors' roles and lifespans, discussions within collaborative KG communities remain understudied. To fill this gap, we investigated the behavioural patterns of contributors and factors affecting their communication and participation. We analysed all the discussions on Wikidata using a mixed methods approach, including statistical tests, network analysis, and text and graph embedding representations. Our findings reveal that the interactions between Wikidata editors form a small world network, resilient to dropouts and inclusive, where both the network topology and discussion content influence the continuity of conversations. Furthermore, the account age of Wikidata members and their conversations are significant factors in their long-term engagement with the project. Our observations and recommendations can benefit the Wikidata and semantic web communities, providing guidance on how to improve collaborative environments for sustainability, growth, and quality.
Cite as

Elisavet Koutsiana, Ioannis Reklos, Kholoud Saad Alghamdi, Nitisha Jain, Albert Meroño-Peñuela, and Elena Simperl. Talking Wikidata: Communication Patterns and Their Impact on Community Engagement in Collaborative Knowledge Graphs. In Transactions on Graph Data and Knowledge (TGDK), Volume 3, Issue 1, pp. 2:1-2:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@Article{koutsiana_et_al:TGDK.3.1.2,
  author =	{Koutsiana, Elisavet and Reklos, Ioannis and Alghamdi, Kholoud Saad and Jain, Nitisha and Mero\~{n}o-Pe\~{n}uela, Albert and Simperl, Elena},
  title =	{{Talking Wikidata: Communication Patterns and Their Impact on Community Engagement in Collaborative Knowledge Graphs}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{2:1--2:27},
  ISSN =	{2942-7517},
  year =	{2025},
  volume =	{3},
  number =	{1},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.3.1.2},
  URN =		{urn:nbn:de:0030-drops-230114},
  doi =		{10.4230/TGDK.3.1.2},
  annote =	{Keywords: collaborative knowledge graph, network analysis, graph embeddings, text embeddings}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.6
On Identity Testing and Noncommutative Rank Computation over the Free Skew Field

Authors: V. Arvind, Abhranil Chatterjee, Utsab Ghosal, Partha Mukhopadhyay, and C. Ramya

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2022.12
Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product

Authors: Arkadev Chattopadhyay, Utsab Ghosal, and Partha Mukhopadhyay

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

Abstract
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.
Cite as

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}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.23
Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time

Authors: V. Arvind, Abhranil Chatterjee, and Partha Mukhopadhyay

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

Abstract
Hrubeš and Wigderson [Hrubeš and Wigderson, 2015] initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box setting, there are deterministic polynomial-time algorithms due to Garg, Gurvits, Oliveira, and Wigderson [Ankit Garg et al., 2016] and Ivanyos, Qiao, and Subrahmanyam [Ivanyos et al., 2018]. A central open problem in this area is to design an efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this for the first nested inverse case. More precisely, we obtain a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative rational formulas of inversion height two via a hitting set construction. Several new technical ideas are involved in the hitting set construction, including concepts from matrix coefficient realization theory [Volčič, 2018] and properties of cyclic division algebras [T.Y. Lam, 2001]. En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas [Michael A. Forbes and Amir Shpilka, 2013] inside a cyclic division algebra of small index.
Cite as

V. Arvind, Abhranil Chatterjee, and Partha Mukhopadhyay. Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{arvind_et_al:LIPIcs.APPROX/RANDOM.2022.23,
  author =	{Arvind, V. and Chatterjee, Abhranil and Mukhopadhyay, Partha},
  title =	{{Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{23:1--23:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.23},
  URN =		{urn:nbn:de:0030-drops-171451},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.23},
  annote =	{Keywords: Rational Identity Testing, Black-box Derandomization, Cyclic Division Algebra, Matrix coefficient realization theory}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.39
Monotone Complexity of Spanning Tree Polynomial Re-Visited

Authors: Arkadev Chattopadhyay, Rajit Datta, Utsab Ghosal, and Partha Mukhopadhyay

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.10
Equivalence Testing of Weighted Automata over Partially Commutative Monoids

Authors: V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
Motivated by equivalence testing of k-tape automata, we study the equivalence testing of weighted automata in the more general setting, over partially commutative monoids (in short, pc monoids), and show efficient algorithms in some special cases, exploiting the structure of the underlying non-commutation graph of the monoid. Specifically, if the edge clique cover number of the non-commutation graph of the pc monoid is a constant, we obtain a deterministic quasi-polynomial time algorithm for equivalence testing. As a corollary, we obtain the first deterministic quasi-polynomial time algorithms for equivalence testing of k-tape weighted automata and for equivalence testing of deterministic k-tape automata for constant k. Prior to this, the best complexity upper bound for these k-tape automata problems were randomized polynomial-time, shown by Worrell [James Worrell, 2013]. Finding a polynomial-time deterministic algorithm for equivalence testing of deterministic k-tape automata for constant k has been open for several years [Emily P. Friedman and Sheila A. Greibach, 1982] and our results make progress. We also consider pc monoids for which the non-commutation graphs have an edge cover consisting of at most k cliques and star graphs for any constant k. We obtain a randomized polynomial-time algorithm for equivalence testing of weighted automata over such monoids. Our results are obtained by designing efficient zero-testing algorithms for weighted automata over such pc monoids.
Cite as

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Equivalence Testing of Weighted Automata over Partially Commutative Monoids. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arvind_et_al:LIPIcs.MFCS.2021.10,
  author =	{Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha},
  title =	{{Equivalence Testing of Weighted Automata over Partially Commutative Monoids}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.10},
  URN =		{urn:nbn:de:0030-drops-144503},
  doi =		{10.4230/LIPIcs.MFCS.2021.10},
  annote =	{Keywords: Weighted Automata, Automata Equivalence, Partially Commutative Monoid}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.36
Parallel Polynomial Permanent Mod Powers of 2 and Shortest Disjoint Cycles

Authors: Samir Datta and Kishlaya Jaiswal

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
We present a parallel algorithm for permanent mod 2^k of a matrix of univariate integer polynomials. It places the problem in ⨁L ⊆ NC². This extends the techniques of Valiant [Leslie G. Valiant, 1979], Braverman, Kulkarni and Roy [Mark Braverman et al., 2009] and Björklund and Husfeldt [Andreas Björklund and Thore Husfeldt, 2019] and yields a (randomized) parallel algorithm for shortest two disjoint paths improving upon the recent (randomized) polynomial time algorithm [Andreas Björklund and Thore Husfeldt, 2019]. We also recognize the disjoint paths problem as a special case of finding disjoint cycles, and present (randomized) parallel algorithms for finding a shortest cycle and shortest two disjoint cycles passing through any given fixed number of vertices or edges.
Cite as

Samir Datta and Kishlaya Jaiswal. Parallel Polynomial Permanent Mod Powers of 2 and Shortest Disjoint Cycles. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 36:1-36:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{datta_et_al:LIPIcs.MFCS.2021.36,
  author =	{Datta, Samir and Jaiswal, Kishlaya},
  title =	{{Parallel Polynomial Permanent Mod Powers of 2 and Shortest Disjoint Cycles}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{36:1--36:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.36},
  URN =		{urn:nbn:de:0030-drops-144763},
  doi =		{10.4230/LIPIcs.MFCS.2021.36},
  annote =	{Keywords: permanent mod powers of 2, parallel computation, graphs, shortest disjoint paths, shortest disjoint cycles}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.10
A Special Case of Rational Identity Testing and the Brešar-Klep Theorem

Authors: V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
We explore a special case of rational identity testing and algorithmic versions of two theorems on noncommutative polynomials, namely, Amitsur's theorem [S.A Amitsur, 1966] and the Brešar-Klep theorem [Brešar and Klep, 2008] when the input polynomial is given by an algebraic branching program (ABP). Let f be a degree-d n-variate noncommutative polynomial in the free ring Q<x_1,x_2,...,x_n> over rationals. 1) We consider the following special case of rational identity testing: Given a noncommutative ABP as white-box, whose edge labels are linear forms or inverses of linear forms, we show a deterministic polynomial-time algorithm to decide if the rational function computed by it is equivalent to zero in the free skew field Q<(X)>. Given black-box access to the ABP, we give a deterministic quasi-polynomial time algorithm for this problem. 2) Amitsur's theorem implies that if a noncommutative polynomial f is nonzero on k x k matrices then, in fact, f(M_1,M_2,...,M_n) is invertible for some matrix tuple (M_1,M_2,...,M_n) in (M_k(ℚ))^n. While a randomized polynomial time algorithm to find such (M_1,M_2,...,M_n) given black-box access to f is simple, we obtain a deterministic s^{O(log d)} time algorithm for the problem with black-box access to f, where s is the minimum ABP size for f and d is the degree of f. 3) The Brešar-Klep Theorem states that the span of the range of any noncommutative polynomial f on k x k matrices over Q is one of the following: zero, scalar multiples of I_k, trace-zero matrices in M_k(Q), or all of M_k(Q). We obtain a deterministic polynomial-time algorithm to decide which case occurs, given white-box access to an ABP for f. We also give a deterministic s^{O(log d)} time algorithm given black-box access to an ABP of size s for f. Our algorithms work when k >= d. Our techniques are based on some automata theory combined with known techniques for noncommutative ABP identity testing [Ran Raz and Amir Shpilka, 2005; Michael A. Forbes and Amir Shpilka, 2013].
Cite as

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. A Special Case of Rational Identity Testing and the Brešar-Klep Theorem. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arvind_et_al:LIPIcs.MFCS.2020.10,
  author =	{Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha},
  title =	{{A Special Case of Rational Identity Testing and the Bre\v{s}ar-Klep Theorem}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.10},
  URN =		{urn:nbn:de:0030-drops-126807},
  doi =		{10.4230/LIPIcs.MFCS.2020.10},
  annote =	{Keywords: Rational identity testing, ABP with inverses, Bre\v{s}ar-Klep Theorem, Invertible image, Amitsur’s theorem}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2019.9
Fast Exact Algorithms Using Hadamard Product of Polynomials

Authors: V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay

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

Abstract
Let C be an arithmetic circuit of poly(n) size given as input that computes a polynomial f in F[X], where X={x_1,x_2,...,x_n} and F is any field where the field arithmetic can be performed efficiently. We obtain new algorithms for the following two problems first studied by Koutis and Williams [Ioannis Koutis, 2008; Ryan Williams, 2009; Ioannis Koutis and Ryan Williams, 2016]. - (k,n)-MLC: Compute the sum of the coefficients of all degree-k multilinear monomials in the polynomial f. - k-MMD: Test if there is a nonzero degree-k multilinear monomial in the polynomial f. Our algorithms are based on the fact that the Hadamard product f o S_{n,k}, is the degree-k multilinear part of f, where S_{n,k} is the k^{th} elementary symmetric polynomial. - For (k,n)-MLC problem, we give a deterministic algorithm of run time O^*(n^(k/2+c log k)) (where c is a constant), answering an open question of Koutis and Williams [Ioannis Koutis and Ryan Williams, 2016]. As corollaries, we show O^*(binom{n}{downarrow k/2})-time exact counting algorithms for several combinatorial problems: k-Tree, t-Dominating Set, m-Dimensional k-Matching. - For k-MMD problem, we give a randomized algorithm of run time 4.32^k * poly(n,k). Our algorithm uses only poly(n,k) space. This matches the run time of a recent algorithm [Cornelius Brand et al., 2018] for k-MMD which requires exponential (in k) space. Other results include fast deterministic algorithms for (k,n)-MLC and k-MMD problems for depth three circuits.
Cite as

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Fast Exact Algorithms Using Hadamard Product of Polynomials. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{arvind_et_al:LIPIcs.FSTTCS.2019.9,
  author =	{Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha},
  title =	{{Fast Exact Algorithms Using Hadamard Product of Polynomials}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{9:1--9:14},
  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.2019.9},
  URN =		{urn:nbn:de:0030-drops-115711},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.9},
  annote =	{Keywords: Hadamard Product, Multilinear Monomial Detection and Counting, Rectangular Permanent, Symmetric Polynomial}
}
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