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Documents authored by Ghosh, Sumanta

Document
DOI: 10.4230/LIPIcs.CCC.2023.2
Border Complexity of Symbolic Determinant Under Rank One Restriction

Authors: Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, and Roshan Raj

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
VBP is the class of polynomial families that can be computed by the determinant of a symbolic matrix of the form A_0 + ∑_{i=1}^n A_i x_i where the size of each A_i is polynomial in the number of variables (equivalently, computable by polynomial-sized algebraic branching programs (ABP)). A major open problem in geometric complexity theory (GCT) is to determine whether VBP is closed under approximation i.e. whether VBP = VBP^ ̅. The power of approximation is well understood for some restricted models of computation, e.g. the class of depth-two circuits, read-once oblivious ABPs (ROABP), monotone ABPs, depth-three circuits of bounded top fan-in, and width-two ABPs. The former three classes are known to be closed under approximation [Markus Bläser et al., 2020], whereas the approximative closure of the last one captures the entire class of polynomial families computable by polynomial-sized formulas [Bringmann et al., 2017]. In this work, we consider the subclass of VBP computed by the determinant of a symbolic matrix of the form A_0 + ∑_{i=1}^n A_i x_i where for each 1 ≤ i ≤ n, A_i is of rank one. This class has been studied extensively [Edmonds, 1968; Jack Edmonds, 1979; Murota, 1993] and efficient identity testing algorithms are known for it [Lovász, 1989; Rohit Gurjar and Thomas Thierauf, 2020]. We show that this class is closed under approximation. In the language of algebraic geometry, we show that the set obtained by taking coordinatewise products of pairs of points from (the Plücker embedding of) a Grassmannian variety is closed.
Cite as

Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, and Roshan Raj. Border Complexity of Symbolic Determinant Under Rank One Restriction. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2023.2,
  author =	{Chatterjee, Abhranil and Ghosh, Sumanta and Gurjar, Rohit and Raj, Roshan},
  title =	{{Border Complexity of Symbolic Determinant Under Rank One Restriction}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{2:1--2:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.2},
  URN =		{urn:nbn:de:0030-drops-182721},
  doi =		{10.4230/LIPIcs.CCC.2023.2},
  annote =	{Keywords: Border Complexity, Symbolic Determinant, Valuated Matroid}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.30
Improved Hitting Set for Orbit of ROABPs

Authors: Vishwas Bhargava and Sumanta Ghosh

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

Abstract
The orbit of an n-variate polynomial f(x) over a field 𝔽 is the set {f(Ax+b) ∣ A ∈ GL(n, 𝔽) and b ∈ 𝔽ⁿ}, and the orbit of a polynomial class is the union of orbits of all the polynomials in it. In this paper, we give improved constructions of hitting-sets for the orbit of read-once oblivious algebraic branching programs (ROABPs) and a related model. Over fields with characteristic zero or greater than d, we construct a hitting set of size (ndw)^{O(w²log n⋅ min{w², dlog w})} for the orbit of ROABPs in unknown variable order where d is the individual degree and w is the width of ROABPs. We also give a hitting set of size (ndw)^{O(min{w²,dlog w})} for the orbit of polynomials computed by w-width ROABPs in any variable order. Our hitting sets improve upon the results of Saha and Thankey [Chandan Saha and Bhargav Thankey, 2021] who gave an (ndw)^{O(dlog w)} size hitting set for the orbit of commutative ROABPs (a subclass of any-order ROABPs) and (nw)^{O(w⁶log n)} size hitting set for the orbit of multilinear ROABPs. Designing better hitting sets in large individual degree regime, for instance d > n, was asked as an open problem by [Chandan Saha and Bhargav Thankey, 2021] and this work solves it in small width setting. We prove some new rank concentration results by establishing low-cone concentration for the polynomials over vector spaces, and they strengthen some previously known low-support based rank concentrations shown in [Michael A. Forbes et al., 2013]. These new low-cone concentration results are crucial in our hitting set construction, and may be of independent interest. To the best of our knowledge, this is the first time when low-cone rank concentration has been used for designing hitting sets.
Cite as

Vishwas Bhargava and Sumanta Ghosh. Improved Hitting Set for Orbit of ROABPs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 30:1-30:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhargava_et_al:LIPIcs.APPROX/RANDOM.2021.30,
  author =	{Bhargava, Vishwas and Ghosh, Sumanta},
  title =	{{Improved Hitting Set for Orbit of ROABPs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{30:1--30:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.30},
  URN =		{urn:nbn:de:0030-drops-147231},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.30},
  annote =	{Keywords: Hitting Set, Low Cone Concentration, Orbits, PIT, ROABP}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.41
Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision

Authors: Sumanta Ghosh and Rohit Gurjar

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

Abstract
We study the matroid intersection problem from the parallel complexity perspective. Given two matroids over the same ground set, the problem asks to decide whether they have a common base and its search version asks to find a common base, if one exists. Another widely studied variant is the weighted decision version where with the two matroids, we are given small weights on the ground set elements and a target weight W, and the question is to decide whether there is a common base of weight at least W. From the perspective of parallel complexity, the relation between the search and the decision versions is not well understood. We make a significant progress on this question by giving a pseudo-deterministic parallel (NC) algorithm for the search version that uses an oracle access to the weighted decision. The notion of pseudo-deterministic NC was recently introduced by Goldwasser and Grossman [Shafi Goldwasser and Ofer Grossman, 2017], which is a relaxation of NC. A pseudo-deterministic NC algorithm for a search problem is a randomized NC algorithm that, for a given input, outputs a fixed solution with high probability. In case the given matroids are linearly representable, our result implies a pseudo-deterministic NC algorithm (without the weighted decision oracle). This resolves an open question posed by Anari and Vazirani [Nima Anari and Vijay V. Vazirani, 2020].
Cite as

Sumanta Ghosh and Rohit Gurjar. Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ghosh_et_al:LIPIcs.APPROX/RANDOM.2021.41,
  author =	{Ghosh, Sumanta and Gurjar, Rohit},
  title =	{{Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{41:1--41:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.41},
  URN =		{urn:nbn:de:0030-drops-147342},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.41},
  annote =	{Keywords: Linear Matroid, Matroid Intersection, Parallel Complexity, Pseudo-deterministic NC}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.54
Towards Blackbox Identity Testing of Log-Variate Circuits

Authors: Michael A. Forbes, Sumanta Ghosh, and Nitin Saxena

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract
Derandomization of blackbox identity testing reduces to extremely special circuit models. After a line of work, it is known that focusing on circuits with constant-depth and constantly many variables is enough (Agrawal,Ghosh,Saxena, STOC'18) to get to general hitting-sets and circuit lower bounds. This inspires us to study circuits with few variables, eg. logarithmic in the size s. We give the first poly(s)-time blackbox identity test for n=O(log s) variate size-s circuits that have poly(s)-dimensional partial derivative space; eg. depth-3 diagonal circuits (or Sigma wedge Sigma^n). The former model is well-studied (Nisan,Wigderson, FOCS'95) but no poly(s2^n)-time identity test was known before us. We introduce the concept of cone-closed basis isolation and prove its usefulness in studying log-variate circuits. It subsumes the previous notions of rank-concentration studied extensively in the context of ROABP models.
Cite as

Michael A. Forbes, Sumanta Ghosh, and Nitin Saxena. Towards Blackbox Identity Testing of Log-Variate Circuits. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{forbes_et_al:LIPIcs.ICALP.2018.54,
  author =	{Forbes, Michael A. and Ghosh, Sumanta and Saxena, Nitin},
  title =	{{Towards Blackbox Identity Testing of Log-Variate Circuits}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{54:1--54:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.54},
  URN =		{urn:nbn:de:0030-drops-90582},
  doi =		{10.4230/LIPIcs.ICALP.2018.54},
  annote =	{Keywords: hitting-set, depth-3, diagonal, derandomization, polynomial identity testing, log-variate, concentration, cone closed, basis isolation}
}
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