13 Search Results for "Kush, Deepanshu"

Document
DOI: 10.4230/LIPIcs.ITCS.2026.46
Symmetric Algebraic Circuits and Homomorphism Polynomials

Authors: Anuj Dawar, Benedikt Pago, and Tim Seppelt

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

Abstract
The central open question of algebraic complexity is whether VP ≠ VNP, which is saying that the permanent cannot be represented by families of polynomial-size algebraic circuits. For symmetric algebraic circuits, this has been confirmed by Dawar and Wilsenach (2020), who showed exponential lower bounds on the size of symmetric circuits for the permanent. In this work, we set out to develop a more general symmetric algebraic complexity theory. Our main result is that a family of symmetric polynomials admits small symmetric circuits if and only if they can be written as a linear combination of homomorphism counting polynomials of graphs of bounded treewidth. We also establish a relationship between the symmetric complexity of subgraph counting polynomials and the vertex cover number of the pattern graph. As a concrete example, we examine the symmetric complexity of immanant families (a generalisation of the determinant and permanent) and show that a known conditional dichotomy due to Curticapean (2021) holds unconditionally in the symmetric setting.
Cite as

Anuj Dawar, Benedikt Pago, and Tim Seppelt. Symmetric Algebraic Circuits and Homomorphism Polynomials. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dawar_et_al:LIPIcs.ITCS.2026.46,
  author =	{Dawar, Anuj and Pago, Benedikt and Seppelt, Tim},
  title =	{{Symmetric Algebraic Circuits and Homomorphism Polynomials}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{46:1--46:15},
  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.2026.46},
  URN =		{urn:nbn:de:0030-drops-253330},
  doi =		{10.4230/LIPIcs.ITCS.2026.46},
  annote =	{Keywords: algebraic complexity, finite model theory, symmetric circuits, homomorphism counting, graph homomorphism, treewidth, counting width, first-order logic with counting quantifiers}
}
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.
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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.IPEC.2025.17
Treedepth Inapproximability and Exponential ETH Lower Bound

Authors: Édouard Bonnet, Daniel Neuen, and Marek Sokołowski

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a 2^{O(k²)} n-time exact algorithm and a polynomial-time O(OPT log^{3/2} OPT)-approximation algorithm, where the former algorithm returns an elimination forest of height k (witnessing that treedepth is at most k) for the n-vertex input graph G, or correctly reports that G has treedepth larger than k, and OPT is the actual value of the treedepth. On the complexity side, exactly computing treedepth is NP-complete, but the known reductions do not rule out a polynomial-time approximation scheme (PTAS), and under the Exponential Time Hypothesis (ETH) only exclude a running time of 2^o(√n) for exact algorithms. We show that 1.0003-approximating Treedepth is NP-hard, and that exactly computing the treedepth of an n-vertex graph requires time 2^Ω(n), unless the ETH fails. We further derive that there exist absolute constants δ, c > 0 such that any (1+δ)-approximation algorithm requires time 2^Ω(n/log^c n). We do so via a simple direct reduction from Satisfiability to Treedepth, inspired by a reduction recently designed for Treewidth [STOC '25].
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Édouard Bonnet, Daniel Neuen, and Marek Sokołowski. Treedepth Inapproximability and Exponential ETH Lower Bound. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 17:1-17:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bonnet_et_al:LIPIcs.IPEC.2025.17,
  author =	{Bonnet, \'{E}douard and Neuen, Daniel and Soko{\l}owski, Marek},
  title =	{{Treedepth Inapproximability and Exponential ETH Lower Bound}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{17:1--17:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.17},
  URN =		{urn:nbn:de:0030-drops-251494},
  doi =		{10.4230/LIPIcs.IPEC.2025.17},
  annote =	{Keywords: treedepth, lower bounds, 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.
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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
DOI: 10.4230/LIPIcs.MFCS.2025.67
#SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank

Authors: Nutan Limaye, Adarsh Srinivasan, and Srikanth Srinivasan

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

Abstract
There is a large body of work that shows how to leverage lower bound techniques for circuit classes to obtain satisfiability algorithms that run in better than brute-force time [Ramamohan Paturi et al., 1997; Ryan Williams, 2014]. For circuits with threshold gates, there are several such algorithms based on either - Probabilistic Representations by low-degree polynomials, which allow for the use of fast polynomial evaluation algorithms, or - Low rank, which allows for an efficient reduction to rectangular matrix multiplication. In this paper, we use a related notion of probabilistic rank to obtain satisfiability algorithms for circuit classes contained in ACC⁰∘3-PTF, i.e. constant-depth circuits with modular counting gates and a single layer of degree-3 polynomial threshold functions. Even for the special case of a single 3-PTF, it is not clear how to use either of the above two strategies to get a non-trivial satisfiability algorithm. The best known algorithm in this case previously was based on memoization and yields worse guarantees than our algorithm.
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Nutan Limaye, Adarsh Srinivasan, and Srikanth Srinivasan. #SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 67:1-67:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{limaye_et_al:LIPIcs.MFCS.2025.67,
  author =	{Limaye, Nutan and Srinivasan, Adarsh and Srinivasan, Srikanth},
  title =	{{#SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{67:1--67:18},
  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.2025.67},
  URN =		{urn:nbn:de:0030-drops-241744},
  doi =		{10.4230/LIPIcs.MFCS.2025.67},
  annote =	{Keywords: probabilistic polynomials, probabilistic rank, circuit satisfiability, circuit lower bounds, polynomial method, threshold circuits}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.15
Algebraic Pseudorandomness in VNC⁰

Authors: Robert Andrews

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

Abstract
We study the arithmetic complexity of hitting set generators, which are pseudorandom objects used for derandomization of the polynomial identity testing problem. We give new explicit constructions of hitting set generators whose outputs are computable in VNC⁰, i.e., can be computed by arithmetic formulas of constant size. Unconditionally, we construct a VNC⁰-computable generator that hits arithmetic circuits of constant depth and polynomial size. We also give conditional constructions, under strong but plausible hardness assumptions, of VNC⁰-computable generators that hit arithmetic formulas and arithmetic branching programs of polynomial size, respectively. As a corollary of our constructions, we derive lower bounds for subsystems of the Geometric Ideal Proof System of Grochow and Pitassi. Constructions of such generators are implicit in prior work of Kayal on lower bounds for the degree of annihilating polynomials. Our main contribution is a construction whose correctness relies on circuit complexity lower bounds rather than degree lower bounds.
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Robert Andrews. Algebraic Pseudorandomness in VNC⁰. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{andrews:LIPIcs.CCC.2025.15,
  author =	{Andrews, Robert},
  title =	{{Algebraic Pseudorandomness in VNC⁰}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{15:1--15:15},
  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.2025.15},
  URN =		{urn:nbn:de:0030-drops-237092},
  doi =		{10.4230/LIPIcs.CCC.2025.15},
  annote =	{Keywords: Polynomial identity testing, Algebraic circuits, Ideal Proof System}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.66
Lipschitz Decompositions of Finite 𝓁_{p} Metrics

Authors: Robert Krauthgamer and Nir Petruschka

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for n-point subsets of 𝓁_p, for p > 2, remained open, see e.g. [Naor, SODA 2017]. We make significant progress on this question and establish the bound β = O(log^{1-1/p} n). Building on prior work, we demonstrate applications of this result to two problems, high-dimensional geometric spanners and distance labeling schemes. In addition, we sharpen a related decomposition bound for 1 < p < 2, due to Filtser and Neiman [Algorithmica 2022].
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Robert Krauthgamer and Nir Petruschka. Lipschitz Decompositions of Finite 𝓁_{p} Metrics. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{krauthgamer_et_al:LIPIcs.SoCG.2025.66,
  author =	{Krauthgamer, Robert and Petruschka, Nir},
  title =	{{Lipschitz Decompositions of Finite 𝓁\underline\{p\} Metrics}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{66:1--66:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.66},
  URN =		{urn:nbn:de:0030-drops-232182},
  doi =		{10.4230/LIPIcs.SoCG.2025.66},
  annote =	{Keywords: Lipschitz decompositions, metric embeddings, geometric spanners}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.6
Equi-Rank Homomorphism Preservation Theorem on Finite Structures

Authors: Benjamin Rossman

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
The Homomorphism Preservation Theorem (HPT) of classical model theory states that a first-order sentence is preserved under homomorphisms if, and only if, it is equivalent to an existential-positive sentence. This theorem remains valid when restricted to finite structures, as demonstrated by the author in [Rossman, 2008; Rossman, 2017] via distinct model-theoretic and circuit-complexity based proofs. In this paper, we present a third (and significantly simpler) proof of the finitary HPT based on a generalized Cai-Fürer-Immerman construction. This method establishes a tight correspondence between syntactic parameters of a homomorphism-preserved sentence (quantifier rank, variable width, alternation height) and structural parameters of its minimal models (tree-width, tree-depth, decomposition height). Consequently, we prove a conjectured "equi-rank" version of the finitary HPT. In contrast, previous versions of the finitary HPT possess additional properties, but incur blow-ups in the quantifier rank of the equivalent existential-positive sentence.
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Benjamin Rossman. Equi-Rank Homomorphism Preservation Theorem on Finite Structures. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{rossman:LIPIcs.CSL.2025.6,
  author =	{Rossman, Benjamin},
  title =	{{Equi-Rank Homomorphism Preservation Theorem on Finite Structures}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.6},
  URN =		{urn:nbn:de:0030-drops-227634},
  doi =		{10.4230/LIPIcs.CSL.2025.6},
  annote =	{Keywords: finite model theory, preservation theorems, quantifier rank}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.20
Lower Bounds for Set-Multilinear Branching Programs

Authors: Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, and Amir Shpilka

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

Abstract
In this paper, we prove super-polynomial lower bounds for the model of sum of ordered set-multilinear algebraic branching programs, each with a possibly different ordering (∑smABP). Specifically, we give an explicit nd-variate polynomial of degree d such that any ∑smABP computing it must have size n^ω(1) for d as low as ω(log n). Notably, this constitutes the first such lower bound in the low degree regime. Moreover, for d = poly(n), we demonstrate an exponential lower bound. This result generalizes the seminal work of Nisan (STOC, 1991), which proved an exponential lower bound for a single ordered set-multilinear ABP. The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (TAMC, 2024), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs - for a polynomial of sufficiently low degree - would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant’s longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by O(log n/ log log n), then it would imply super-polynomial lower bounds against general ABPs. Our results strengthen the works of Arvind & Raja (Chic. J. Theor. Comput. Sci., 2016) and Bhargav, Dwivedi & Saxena (TAMC, 2024), as well as the works of Ramya & Rao (Theor. Comput. Sci., 2020) and Ghoshal & Rao (International Computer Science Symposium in Russia, 2021), each of which established lower bounds for related or restricted versions of this model. They also strongly answer a question from the former two, which asked to prove super-polynomial lower bounds for general ∑smABP.
Cite as

Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, and Amir Shpilka. Lower Bounds for Set-Multilinear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2024.20,
  author =	{Chatterjee, Prerona and Kush, Deepanshu and Saraf, Shubhangi and Shpilka, Amir},
  title =	{{Lower Bounds for Set-Multilinear Branching Programs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.20},
  URN =		{urn:nbn:de:0030-drops-204167},
  doi =		{10.4230/LIPIcs.CCC.2024.20},
  annote =	{Keywords: Lower Bounds, Algebraic Branching Programs, Set-multilinear polynomials}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.15
Near-Optimal Set-Multilinear Formula Lower Bounds

Authors: Deepanshu Kush and Shubhangi Saraf

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

Abstract
The seminal work of Raz (J. ACM 2013) as well as the recent breakthrough results by Limaye, Srinivasan, and Tavenas (FOCS 2021, STOC 2022) have demonstrated a potential avenue for obtaining lower bounds for general algebraic formulas, via strong enough lower bounds for set-multilinear formulas. In this paper, we make progress along this direction by proving near-optimal lower bounds against low-depth as well as unbounded-depth set-multilinear formulas. More precisely, we show that over any field of characteristic zero, there is a polynomial f computed by a polynomial-sized set-multilinear branching program (i.e., f is in set-multilinear VBP) defined over Θ(n²) variables and of degree Θ(n), such that any product-depth Δ set-multilinear formula computing f has size at least n^Ω(n^{1/Δ}/Δ). Moreover, we show that any unbounded-depth set-multilinear formula computing f has size at least n^{Ω(log n)}. If such strong lower bounds are proven for the iterated matrix multiplication (IMM) polynomial or rather, any polynomial that is computed by an ordered set-multilinear branching program (i.e., a further restriction of set-multilinear VBP), then this would have dramatic consequences as it would imply super-polynomial lower bounds for general algebraic formulas (Raz, J. ACM 2013; Tavenas, Limaye, and Srinivasan, STOC 2022). Prior to our work, either only weaker lower bounds were known for the IMM polynomial (Tavenas, Limaye, and Srinivasan, STOC 2022), or similar strong lower bounds were known but for a hard polynomial not known to be even in set-multilinear VP (Kush and Saraf, CCC 2022; Raz, J. ACM 2009). By known depth-reduction results, our lower bounds are essentially tight for f and in general, for any hard polynomial that is in set-multilinear VBP or set-multilinear VP. Any asymptotic improvement in the lower bound (for a hard polynomial, say, in VNP) would imply super-polynomial lower bounds for general set-multilinear circuits.
Cite as

Deepanshu Kush and Shubhangi Saraf. Near-Optimal Set-Multilinear Formula Lower Bounds. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 15:1-15:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kush_et_al:LIPIcs.CCC.2023.15,
  author =	{Kush, Deepanshu and Saraf, Shubhangi},
  title =	{{Near-Optimal Set-Multilinear Formula Lower Bounds}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{15:1--15:33},
  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.2023.15},
  URN =		{urn:nbn:de:0030-drops-182855},
  doi =		{10.4230/LIPIcs.CCC.2023.15},
  annote =	{Keywords: Algebraic Complexity, Set-multilinear, Formula Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.38
Improved Low-Depth Set-Multilinear Circuit Lower Bounds

Authors: Deepanshu Kush and Shubhangi Saraf

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract
In this paper, we prove strengthened lower bounds for constant-depth set-multilinear formulas. More precisely, we show that over any field, there is an explicit polynomial f in VNP defined over n² variables, and of degree n, such that any product-depth Δ set-multilinear formula computing f has size at least n^Ω(n^{1/Δ}/Δ). The hard polynomial f comes from the class of Nisan-Wigderson (NW) design-based polynomials. Our lower bounds improve upon the recent work of Limaye, Srinivasan and Tavenas (STOC 2022), where a lower bound of the form (log n)^Ω(Δ n^{1/Δ}) was shown for the size of product-depth Δ set-multilinear formulas computing the iterated matrix multiplication (IMM) polynomial of the same degree and over the same number of variables as f. Moreover, our lower bounds are novel for any Δ ≥ 2. The precise quantitative expression in our lower bound is interesting also because the lower bounds we obtain are "sharp" in the sense that any asymptotic improvement would imply general set-multilinear circuit lower bounds via depth reduction results. In the setting of general set-multilinear formulas, a lower bound of the form n^Ω(log n) was already obtained by Raz (J. ACM 2009) for the more general model of multilinear formulas. The techniques of LST (which extend the techniques of the same authors in (FOCS 2021)) give a different route to set-multilinear formula lower bounds, and allow them to obtain a lower bound of the form (log n)^Ω(log n) for the size of general set-multilinear formulas computing the IMM polynomial. Our proof techniques are another variation on those of LST, and enable us to show an improved lower bound (matching that of Raz) of the form n^Ω(log n), albeit for the same polynomial f in VNP (the NW polynomial). As observed by LST, if the same n^Ω(log n) size lower bounds for unbounded-depth set-multilinear formulas could be obtained for the IMM polynomial, then using the self-reducibility of IMM and using hardness escalation results, this would imply super-polynomial lower bounds for general algebraic formulas.
Cite as

Deepanshu Kush and Shubhangi Saraf. Improved Low-Depth Set-Multilinear Circuit Lower Bounds. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kush_et_al:LIPIcs.CCC.2022.38,
  author =	{Kush, Deepanshu and Saraf, Shubhangi},
  title =	{{Improved Low-Depth Set-Multilinear Circuit Lower Bounds}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{38:1--38:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.38},
  URN =		{urn:nbn:de:0030-drops-166003},
  doi =		{10.4230/LIPIcs.CCC.2022.38},
  annote =	{Keywords: algebraic circuit complexity, complexity measure, set-multilinear formulas}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.50
Near Neighbor Search via Efficient Average Distortion Embeddings

Authors: Deepanshu Kush, Aleksandar Nikolov, and Haohua Tang

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
A recent series of papers by Andoni, Naor, Nikolov, Razenshteyn, and Waingarten (STOC 2018, FOCS 2018) has given approximate near neighbour search (NNS) data structures for a wide class of distance metrics, including all norms. In particular, these data structures achieve approximation on the order of p for 𝓁_p^d norms with space complexity nearly linear in the dataset size n and polynomial in the dimension d, and query time sub-linear in n and polynomial in d. The main shortcoming is the exponential in d pre-processing time required for their construction. In this paper, we describe a more direct framework for constructing NNS data structures for general norms. More specifically, we show via an algorithmic reduction that an efficient NNS data structure for a metric ℳ is implied by an efficient average distortion embedding of ℳ into 𝓁₁ or the Euclidean space. In particular, the resulting data structures require only polynomial pre-processing time, as long as the embedding can be computed in polynomial time. As a concrete instantiation of this framework, we give an NNS data structure for 𝓁_p with efficient pre-processing that matches the approximation factor, space and query complexity of the aforementioned data structure of Andoni et al. On the way, we resolve a question of Naor (Analysis and Geometry in Metric Spaces, 2014) and provide an explicit, efficiently computable embedding of 𝓁_p, for p ≥ 1, into 𝓁₁ with average distortion on the order of p. Furthermore, we also give data structures for Schatten-p spaces with improved space and query complexity, albeit still requiring exponential pre-processing when p ≥ 2. We expect our approach to pave the way for constructing efficient NNS data structures for all norms.
Cite as

Deepanshu Kush, Aleksandar Nikolov, and Haohua Tang. Near Neighbor Search via Efficient Average Distortion Embeddings. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kush_et_al:LIPIcs.SoCG.2021.50,
  author =	{Kush, Deepanshu and Nikolov, Aleksandar and Tang, Haohua},
  title =	{{Near Neighbor Search via Efficient Average Distortion Embeddings}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{50:1--50:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.50},
  URN =		{urn:nbn:de:0030-drops-138490},
  doi =		{10.4230/LIPIcs.SoCG.2021.50},
  annote =	{Keywords: Nearest neighbor search, metric space embeddings, average distortion embeddings, locality-sensitive hashing}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2019.8
A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates

Authors: Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, Nutan Limaye, and Srikanth Srinivasan

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract
We show that there is a zero-error randomized algorithm that, when given a small constant-depth Boolean circuit C made up of gates that compute constant-degree Polynomial Threshold functions or PTFs (i.e., Boolean functions that compute signs of constant-degree polynomials), counts the number of satisfying assignments to C in significantly better than brute-force time. Formally, for any constants d,k, there is an epsilon > 0 such that the zero-error randomized algorithm counts the number of satisfying assignments to a given depth-d circuit C made up of k-PTF gates such that C has size at most n^{1+epsilon}. The algorithm runs in time 2^{n-n^{Omega(epsilon)}}. Before our result, no algorithm for beating brute-force search was known for counting the number of satisfying assignments even for a single degree-k PTF (which is a depth-1 circuit of linear size). The main new tool is the use of a learning algorithm for learning degree-1 PTFs (or Linear Threshold Functions) using comparison queries due to Kane, Lovett, Moran and Zhang (FOCS 2017). We show that their ideas fit nicely into a memoization approach that yields the #SAT algorithms.
Cite as

Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, Nutan Limaye, and Srikanth Srinivasan. A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bajpai_et_al:LIPIcs.ITCS.2019.8,
  author =	{Bajpai, Swapnam and Krishan, Vaibhav and Kush, Deepanshu and Limaye, Nutan and Srinivasan, Srikanth},
  title =	{{A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.8},
  URN =		{urn:nbn:de:0030-drops-101010},
  doi =		{10.4230/LIPIcs.ITCS.2019.8},
  annote =	{Keywords: SAT, Polynomial Threshold Functions, Constant-depth Boolean Circuits, Linear Decision Trees, Zero-error randomized algorithms}
}
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