9 Search Results for "Thankey, Bhargav"

Document
DOI: 10.4230/LIPIcs.STACS.2026.28
Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank

Authors: Nikolai Chukhin, Alexander S. Kulikov, Ivan Mihajlin, and Arina Smirnova

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
Proving complexity lower bounds remains a challenging task: currently, we only know how to prove conditional uniform (algorithm) lower bounds and nonuniform (circuit) lower bounds in restricted circuit models. About a decade ago, Williams (STOC 2010) showed how to derive nonuniform lower bounds from uniform upper bounds: roughly, by designing a fast algorithm for checking satisfiability of circuits, one gets a lower bound for this circuit class. Since then, a number of results of this kind have been proved. For example, Jahanjou et al. (ICALP 2015) and Carmosino et al. (ITCS 2016) proved that if NSETH fails, then E^{NP} has series-parallel circuit size ω(n). One can also derive nonuniform lower bounds from nondeterministic uniform lower bounds. Perhaps the most well-known example is the Karp-Lipton theorem (STOC 1980): if Σ₂ ≠ Π₂, then NP ⊄ P/poly. Some recent examples include the following. Nederlof (STOC 2020) proved a lower bound on the matrix multiplication tensor rank under an assumption that TSP cannot be solved faster than in 2ⁿ time. Belova et al. (SODA 2024) proved that there exists an explicit polynomial family of arithmetic circuit size Ω(n^{δ}), for any δ > 0, assuming that MAX-3-SAT cannot be solved faster than in 2ⁿ nondeterministic time. Williams (FOCS 2024) proved an exponential lower bound for ETHR ∘ ETHR circuits under the Orthogonal Vectors conjecture. Whereas all the lower bounds above are proved under strong assumptions that might eventually be refuted, the revealed connections are of great interest and may still give further insights: one may be able to weaken the used assumptions or to construct generators from other fine-grained reductions. In this paper, we continue developing this line of research and show how uniform nondeterministic lower bounds can be used to construct generators of various types of combinatorial objects that are notoriously hard to analyze: Boolean functions of high circuit size, matrices of high rigidity, and tensors of high rank. Specifically, we prove the following. - If, for some ε and k, k-SAT cannot be solved in input-oblivious co-nondeterministic time O(2^{(1/2+ε)n}), then there exists a monotone Boolean function family in coNP of monotone circuit size 2^{Ω(n / log n)}. Combining this with the result above, we get win-win circuit lower bounds: either E^{NP{}} requires series-parallel circuits of size ω(n) or coNP requires monotone circuits of size 2^{Ω(n / log n)}. - If, for all ε > 0, MAX-3-SAT cannot be solved in co-nondeterministic time O(2^{(1 - ε)n}), then there exist small families of matrices with rigidity exceeding the best known constructions as well as small families of three-dimensional tensors of rank n^{1+Δ}, for some Δ > 0.
Cite as

Nikolai Chukhin, Alexander S. Kulikov, Ivan Mihajlin, and Arina Smirnova. Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chukhin_et_al:LIPIcs.STACS.2026.28,
  author =	{Chukhin, Nikolai and Kulikov, Alexander S. and Mihajlin, Ivan and Smirnova, Arina},
  title =	{{Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{28:1--28:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.28},
  URN =		{urn:nbn:de:0030-drops-255177},
  doi =		{10.4230/LIPIcs.STACS.2026.28},
  annote =	{Keywords: computational complexity, circuit complexity, lower bounds, conditional lower bounds, monotone circuits, matrix rigidity, tensor rank, arithmetic circuits, fine-grained complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.9
On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

Authors: Robert Andrews, Jules Armand, Prateek Dwivedi, Magnus Rahbek Dalgaard Hansen, Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas

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

Abstract
We investigate the closure properties of read-once oblivious Algebraic Branching Programs (roABPs) under various natural algebraic operations and prove the following. - Non-closure under factoring: There is a sequence of explicit polynomials (f_n(x₁,…, x_n))_n that have poly(n)-sized roABPs such that some irreducible factor of f_n requires roABPs of superpolynomial size in any order. - Non-closure under powering: There is a sequence of polynomials (f_n(x₁,…, x_n))_n with poly(n)-sized roABPs such that any super-constant power of f_n does not have roABPs of polynomial size in any order (and f_nⁿ requires exponential size in any order). - Non-closure under symmetric operations: There are symmetric polynomials (f_n(e₁,…, e_n))_n that have roABPs of polynomial size such that f_n(x₁,…, x_n) do not have roABPs of subexponential size. (Here, e₁,…, e_n denote the elementary symmetric polynomials in n variables.) These results should be viewed in light of known results on models such as algebraic circuits, (general) algebraic branching programs, formulas and constant-depth circuits, all of which are known to be closed under these operations. To prove non-closure under factoring, we construct hard polynomials based on expander graphs using gadgets that lift their hardness from sparse polynomials to roABPs. For symmetric compositions, we show that the circulant polynomial requires roABPs of exponential size in every variable order.
Cite as

Robert Andrews, Jules Armand, Prateek Dwivedi, Magnus Rahbek Dalgaard Hansen, Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. On Closure Properties of Read-Once Oblivious Algebraic Branching Programs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{andrews_et_al:LIPIcs.ITCS.2026.9,
  author =	{Andrews, Robert and Armand, Jules and Dwivedi, Prateek and Hansen, Magnus Rahbek Dalgaard and Limaye, Nutan and Srinivasan, Srikanth and Tavenas, S\'{e}bastien},
  title =	{{On Closure Properties of Read-Once Oblivious Algebraic Branching Programs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{9:1--9:21},
  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.2026.9},
  URN =		{urn:nbn:de:0030-drops-252964},
  doi =		{10.4230/LIPIcs.ITCS.2026.9},
  annote =	{Keywords: Factoring, Closure Properties, Sparsity Bounds, Symmetric Polynomials, roABP, Expander Graphs}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.113
Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP

Authors: Benjamin Rossman and Davidson Zhu

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

Abstract
The sum-of-squares (SoS) complexity of a d-multiquadratic polynomial f (quadratic in each of d blocks of n variables) is the minimum s such that f = ∑_{i = 1}^s g_i² with each g_i d-multilinear. In the case d = 2, Hrubeš, Wigderson and Yehudayoff [Hrubeš et al., 2011] showed that an n^{1+Ω(1)} lower bound on the SoS complexity of explicit biquadratic polynomials implies an exponential lower bound for non-commutative arithmetic circuits. In this paper, we establish an analogous connection between general multiquadratic sum-of-squares and commutative arithmetic formulas. Specifically, we show that an n^{d-o(log d)} lower bound on the SoS complexity of explicit d-multiquadratic polynomials, for any d = d(n) with ω(1) ≤ d(n) ≤ O((log n)/(log log n)), would separate the algebraic complexity classes VNC¹ and VNP.
Cite as

Benjamin Rossman and Davidson Zhu. Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 113:1-113:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{rossman_et_al:LIPIcs.ITCS.2026.113,
  author =	{Rossman, Benjamin and Zhu, Davidson},
  title =	{{Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{113:1--113:22},
  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.2026.113},
  URN =		{urn:nbn:de:0030-drops-254006},
  doi =		{10.4230/LIPIcs.ITCS.2026.113},
  annote =	{Keywords: sum-of-squares, arithmetic formulas}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2025.49
On the Hardness of Order Finding and Equivalence Testing for ROABPs

Authors: C. Ramya and Pratik Shastri

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

Abstract
The complexity of representing a polynomial by a Read-Once Oblivious Algebraic Branching Program (ROABP) is highly dependent on the chosen variable ordering. Bhargava et al. [Bhargava et al., 2024] prove that finding the optimal ordering is NP-hard, and provide some evidence (based on the Small Set Expansion hypothesis) that it is also hard to approximate the optimal ROABP width. In another work, Baraskar et al. [Baraskar et al., 2024] show that it is NP-hard to test whether a polynomial is in the GL_n orbit of a polynomial of sparsity at most s. Building upon these works, we show the following results: first, we prove that approximating the minimum ROABP width up to any constant factor is NP-hard, when the input is presented as a circuit. This removes the reliance on stronger conjectures in the previous work [Bhargava et al., 2024]. Second, we show that testing if an input polynomial given in the sparse representation is in the affine GL_n orbit of a width-w ROABP is NP-hard. Furthermore, we show that over fields of characteristic 0, the problem is NP-hard even when the input polynomial is homogeneous. This provides the first NP-hardness results for membership testing for a dense subclass of polynomial sized algebraic branching programs (VBP). Finally, we locate the source of hardness for the order finding problem at the lowest possible non-trivial degree, proving that the problem is NP-hard even for quadratic forms.
Cite as

C. Ramya and Pratik Shastri. On the Hardness of Order Finding and Equivalence Testing for 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. 49:1-49:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ramya_et_al:LIPIcs.FSTTCS.2025.49,
  author =	{Ramya, C. and Shastri, Pratik},
  title =	{{On the Hardness of Order Finding and Equivalence Testing for ROABPs}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{49:1--49:13},
  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.2025.49},
  URN =		{urn:nbn:de:0030-drops-251296},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.49},
  annote =	{Keywords: ROABP, Order Finding, Equivalence Testing, NP-hardness, Hardness of Approximation}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.19
Monotone Bounded-Depth Complexity of Homomorphism Polynomials

Authors: C.S. Bhargav, Shiteng Chen, Radu Curticapean, and Prateek Dwivedi

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
For every fixed graph H, it is known that homomorphism counts from H and colorful H-subgraph counts can be determined in O(n^{t+1}) time on n-vertex input graphs G, where t is the treewidth of H. On the other hand, a running time of n^{o(t / log t)} would refute the exponential-time hypothesis. Komarath, Pandey, and Rahul (Algorithmica, 2023) studied algebraic variants of these counting problems, i.e., homomorphism and subgraph polynomials for fixed graphs H. These polynomials are weighted sums over the objects counted above, where each object is weighted by the product of variables corresponding to edges contained in the object. As shown by Komarath et al., the monotone circuit complexity of the homomorphism polynomial for H is Θ(n^{tw(H)+1}). In this paper, we characterize the power of monotone bounded-depth circuits for homomorphism and colorful subgraph polynomials. This leads us to discover a natural hierarchy of graph parameters tw_Δ(H), for fixed Δ ∈ ℕ, which capture the width of tree-decompositions for H when the underlying tree is required to have depth at most Δ. We prove that monotone circuits of product-depth Δ computing the homomorphism polynomial for H require size Θ(n^{tw_Δ(H^{†})+1}), where H^{†} is the graph obtained from H by removing all degree-1 vertices. This allows us to derive an optimal depth hierarchy theorem for monotone bounded-depth circuits through graph-theoretic arguments.
Cite as

C.S. Bhargav, Shiteng Chen, Radu Curticapean, and Prateek Dwivedi. Monotone Bounded-Depth Complexity of Homomorphism Polynomials. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhargav_et_al:LIPIcs.MFCS.2025.19,
  author =	{Bhargav, C.S. and Chen, Shiteng and Curticapean, Radu and Dwivedi, Prateek},
  title =	{{Monotone Bounded-Depth Complexity of Homomorphism Polynomials}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{19:1--19:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.19},
  URN =		{urn:nbn:de:0030-drops-241269},
  doi =		{10.4230/LIPIcs.MFCS.2025.19},
  annote =	{Keywords: algebraic complexity, homomorphisms, monotone circuit complexity, bounded-depth circuits, treewidth, pathwidth}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.12
Low-Depth Arithmetic Circuit Lower Bounds: Bypassing Set-Multilinearization

Authors: Prashanth Amireddy, Ankit Garg, Neeraj Kayal, Chandan Saha, and Bhargav Thankey

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
A recent breakthrough work of Limaye, Srinivasan and Tavenas [Nutan Limaye et al., 2021] proved superpolynomial lower bounds for low-depth arithmetic circuits via a "hardness escalation" approach: they proved lower bounds for low-depth set-multilinear circuits and then lifted the bounds to low-depth general circuits. In this work, we prove superpolynomial lower bounds for low-depth circuits by bypassing the hardness escalation, i.e., the set-multilinearization, step. As set-multilinearization comes with an exponential blow-up in circuit size, our direct proof opens up the possibility of proving an exponential lower bound for low-depth homogeneous circuits by evading a crucial bottleneck. Our bounds hold for the iterated matrix multiplication and the Nisan-Wigderson design polynomials. We also define a subclass of unrestricted depth homogeneous formulas which we call unique parse tree (UPT) formulas, and prove superpolynomial lower bounds for these. This significantly generalizes the superpolynomial lower bounds for regular formulas [Neeraj Kayal et al., 2014; Hervé Fournier et al., 2015].
Cite as

Prashanth Amireddy, Ankit Garg, Neeraj Kayal, Chandan Saha, and Bhargav Thankey. Low-Depth Arithmetic Circuit Lower Bounds: Bypassing Set-Multilinearization. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{amireddy_et_al:LIPIcs.ICALP.2023.12,
  author =	{Amireddy, Prashanth and Garg, Ankit and Kayal, Neeraj and Saha, Chandan and Thankey, Bhargav},
  title =	{{Low-Depth Arithmetic Circuit Lower Bounds: Bypassing Set-Multilinearization}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{12:1--12:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.12},
  URN =		{urn:nbn:de:0030-drops-180642},
  doi =		{10.4230/LIPIcs.ICALP.2023.12},
  annote =	{Keywords: arithmetic circuits, low-depth circuits, lower bounds, shifted partials}
}
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.2021.50
Hitting Sets for Orbits of Circuit Classes and Polynomial Families

Authors: Chandan Saha and Bhargav Thankey

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}. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: 1) Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials. 2) Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family {IMM_{3,d}}_{d ∈ ℕ}, which is complete for arithmetic formulas. 3) Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials. 4) Polynomials computable by occur-once formulas.
Cite as

Chandan Saha and Bhargav Thankey. Hitting Sets for Orbits of Circuit Classes and Polynomial Families. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 50:1-50:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{saha_et_al:LIPIcs.APPROX/RANDOM.2021.50,
  author =	{Saha, Chandan and Thankey, Bhargav},
  title =	{{Hitting Sets for Orbits of Circuit Classes and Polynomial Families}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{50:1--50:26},
  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.2021.50},
  URN =		{urn:nbn:de:0030-drops-147433},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.50},
  annote =	{Keywords: Hitting Sets, Orbits, ROABPs, Rank Concentration}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.23
A Super-Quadratic Lower Bound for Depth Four Arithmetic Circuits

Authors: Nikhil Gupta, Chandan Saha, and Bhargav Thankey

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

Abstract
We show an Ω̃(n^2.5) lower bound for general depth four arithmetic circuits computing an explicit n-variate degree-Θ(n) multilinear polynomial over any field of characteristic zero. To our knowledge, and as stated in the survey [Amir Shpilka and Amir Yehudayoff, 2010], no super-quadratic lower bound was known for depth four circuits over fields of characteristic ≠ 2 before this work. The previous best lower bound is Ω̃(n^1.5) [Abhijat Sharma, 2017], which is a slight quantitative improvement over the roughly Ω(n^1.33) bound obtained by invoking the super-linear lower bound for constant depth circuits in [Ran Raz, 2010; Victor Shoup and Roman Smolensky, 1997]. Our lower bound proof follows the approach of the almost cubic lower bound for depth three circuits in [Neeraj Kayal et al., 2016] by replacing the shifted partials measure with a suitable variant of the projected shifted partials measure, but it differs from [Neeraj Kayal et al., 2016]’s proof at a crucial step - namely, the way "heavy" product gates are handled. Loosely speaking, a heavy product gate has a relatively high fan-in. Product gates of a depth three circuit compute products of affine forms, and so, it is easy to prune Θ(n) many heavy product gates by projecting the circuit to a low-dimensional affine subspace [Neeraj Kayal et al., 2016; Amir Shpilka and Avi Wigderson, 2001]. However, in a depth four circuit, the second (from the top) layer of product gates compute products of polynomials having arbitrary degree, and hence it was not clear how to prune such heavy product gates from the circuit. We show that heavy product gates can also be eliminated from a depth four circuit by projecting the circuit to a low-dimensional affine subspace, unless the heavy gates together account for Ω̃(n^2.5) size. This part of our argument is inspired by a well-known greedy approximation algorithm for the weighted set-cover problem.
Cite as

Nikhil Gupta, Chandan Saha, and Bhargav Thankey. A Super-Quadratic Lower Bound for Depth Four Arithmetic Circuits. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 23:1-23:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gupta_et_al:LIPIcs.CCC.2020.23,
  author =	{Gupta, Nikhil and Saha, Chandan and Thankey, Bhargav},
  title =	{{A Super-Quadratic Lower Bound for Depth Four Arithmetic Circuits}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{23:1--23:31},
  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.2020.23},
  URN =		{urn:nbn:de:0030-drops-125757},
  doi =		{10.4230/LIPIcs.CCC.2020.23},
  annote =	{Keywords: depth four arithmetic circuits, Projected Shifted Partials, super-quadratic lower bound}
}
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