19 Search Results for "Grochow, Joshua A."

Document
DOI: 10.4230/LIPIcs.CCC.2024.29
Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity

Authors: Shuichi Hirahara, Valentine Kabanets, Zhenjian Lu, and Igor C. Oliveira

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

Abstract
A search-to-decision reduction is a procedure that allows one to find a solution to a problem from the mere ability to decide when a solution exists. The existence of a search-to-decision reduction for time-bounded Kolmogorov complexity, i.e., the problem of checking if a string x can be generated by a t-time bounded program of description length s, is a long-standing open problem that dates back to the 1960s. In this work, we obtain new average-case and worst-case search-to-decision reductions for the complexity measure 𝖪^t and its randomized analogue rK^t: 1) (Conditional Errorless and Error-Prone Reductions for 𝖪^t) Under the assumption that 𝖤 requires exponential size circuits, we design polynomial-time average-case search-to-decision reductions for 𝖪^t in both errorless and error-prone settings. In fact, under the easiness of deciding 𝖪^t under the uniform distribution, we obtain a search algorithm for any given polynomial-time samplable distribution. In the error-prone reduction, the search algorithm works in the more general setting of conditional 𝖪^t complexity, i.e., it finds a minimum length t-time bound program for generating x given a string y. 2) (Unconditional Errorless Reduction for rK^t) We obtain an unconditional polynomial-time average-case search-to-decision reduction for rK^t in the errorless setting. Similarly to the results described above, we obtain a search algorithm for each polynomial-time samplable distribution, assuming the existence of a decision algorithm under the uniform distribution. To our knowledge, this is the first unconditional sub-exponential time search-to-decision reduction among the measures 𝖪^t and rK^t that works with respect to any given polynomial-time samplable distribution. 3) (Worst-Case to Average-Case Reductions) Under the errorless average-case easiness of deciding rK^t, we design a worst-case search algorithm running in time 2^O(n/log n) that produces a minimum length randomized t-time program for every input string x ∈ {0,1}ⁿ, with the caveat that it only succeeds on some explicitly computed sub-exponential time bound t ≤ 2^{n^ε} that depends on x. A similar result holds for 𝖪^t, under the assumption that 𝖤 requires exponential size circuits. In these results, the corresponding search problem is solved exactly, i.e., a successful run of the search algorithm outputs a t-time bounded program for x of minimum length, as opposed to an approximately optimal program of slightly larger description length or running time.
Cite as

Shuichi Hirahara, Valentine Kabanets, Zhenjian Lu, and Igor C. Oliveira. Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 29:1-29:56, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{hirahara_et_al:LIPIcs.CCC.2024.29,
  author =	{Hirahara, Shuichi and Kabanets, Valentine and Lu, Zhenjian and Oliveira, Igor C.},
  title =	{{Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{29:1--29:56},
  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.2024.29},
  URN =		{urn:nbn:de:0030-drops-204256},
  doi =		{10.4230/LIPIcs.CCC.2024.29},
  annote =	{Keywords: average-case complexity, Kolmogorov complexity, search-to-decision reductions}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.31
Low-Depth Algebraic Circuit Lower Bounds over Any Field

Authors: Michael A. Forbes

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

Abstract
The recent breakthrough of Limaye, Srinivasan and Tavenas [Limaye et al., 2022] (LST) gave the first super-polynomial lower bounds against low-depth algebraic circuits, for any field of zero (or sufficiently large) characteristic. It was an open question to extend this result to small-characteristic ([Limaye et al., 2022; Govindasamy et al., 2022; Fournier et al., 2023]), which in particular is relevant for an approach to prove superpolynomial AC⁰[p]-Frege lower bounds ([Govindasamy et al., 2022]). In this work, we prove super-polynomial algebraic circuit lower bounds against low-depth algebraic circuits over any field, with the same parameters as LST (or even matching the improved parameters of Bhargav, Dutta, and Saxena [Bhargav et al., 2022]). We give two proofs. The first is logical, showing that even though the proof of LST naively fails in small characteristic, the proof is sufficiently algebraic that generic transfer results imply the result over characteristic zero implies the result over all fields. Motivated by this indirect proof, we then proceed to give a second constructive proof, replacing the field-dependent set-multilinearization result of LST with a set-multilinearization that works over any field, by using the Binet-Minc identity [Minc, 1979].
Cite as

Michael A. Forbes. Low-Depth Algebraic Circuit Lower Bounds over Any Field. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{forbes:LIPIcs.CCC.2024.31,
  author =	{Forbes, Michael A.},
  title =	{{Low-Depth Algebraic Circuit Lower Bounds over Any Field}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{31:1--31:16},
  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.2024.31},
  URN =		{urn:nbn:de:0030-drops-204271},
  doi =		{10.4230/LIPIcs.CCC.2024.31},
  annote =	{Keywords: algebraic circuits, lower bounds, low-depth circuits, positive characteristic}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.12
From Proof Complexity to Circuit Complexity via Interactive Protocols

Authors: Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Folklore in complexity theory suspects that circuit lower bounds against NC¹ or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP ⊈ P/poly, as recently observed by Pich and Santhanam [Pich and Santhanam, 2023]. We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Krajíček [Krajíček, 2004], capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies #P ⊈ FP/poly (which would in turn imply, for example, PSPACE ⊈ P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan [Lund et al., 1992]. This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.
Cite as

Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam. From Proof Complexity to Circuit Complexity via Interactive Protocols. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{arteche_et_al:LIPIcs.ICALP.2024.12,
  author =	{Arteche, Noel and Khaniki, Erfan and Pich, J\'{a}n and Santhanam, Rahul},
  title =	{{From Proof Complexity to Circuit Complexity via Interactive Protocols}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{12:1--12:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.12},
  URN =		{urn:nbn:de:0030-drops-201557},
  doi =		{10.4230/LIPIcs.ICALP.2024.12},
  annote =	{Keywords: proof complexity, circuit complexity, interactive protocols}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.117
Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle

Authors: Aaron Potechin and Aaron Zhang

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We show that the minimum total coefficient size of a Nullstellensatz proof of the pigeonhole principle on n+1 pigeons and n holes is 2^{Θ(n)}. We also investigate the ordering principle and construct an explicit Nullstellensatz proof for the ordering principle on n elements with total coefficient size 2ⁿ - n.
Cite as

Aaron Potechin and Aaron Zhang. Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 117:1-117:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{potechin_et_al:LIPIcs.ICALP.2024.117,
  author =	{Potechin, Aaron and Zhang, Aaron},
  title =	{{Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{117:1--117:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.117},
  URN =		{urn:nbn:de:0030-drops-202604},
  doi =		{10.4230/LIPIcs.ICALP.2024.117},
  annote =	{Keywords: Proof complexity, Nullstellensatz, pigeonhole principle, coefficient size}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.31
On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials III: Actions by Classical Groups

Authors: Zhili Chen, Joshua A. Grochow, Youming Qiao, Gang Tang, and Chuanqi Zhang

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
We study the complexity of isomorphism problems for d-way arrays, or tensors, under natural actions by classical groups such as orthogonal, unitary, and symplectic groups. These problems arise naturally in statistical data analysis and quantum information. We study two types of complexity-theoretic questions. First, for a fixed action type (isomorphism, conjugacy, etc.), we relate the complexity of the isomorphism problem over a classical group to that over the general linear group. Second, for a fixed group type (orthogonal, unitary, or symplectic), we compare the complexity of the isomorphism problems for different actions. Our main results are as follows. First, for orthogonal and symplectic groups acting on 3-way arrays, the isomorphism problems reduce to the corresponding problems over the general linear group. Second, for orthogonal and unitary groups, the isomorphism problems of five natural actions on 3-way arrays are polynomial-time equivalent, and the d-tensor isomorphism problem reduces to the 3-tensor isomorphism problem for any fixed d > 3. For unitary groups, the preceding result implies that LOCC classification of tripartite quantum states is at least as difficult as LOCC classification of d-partite quantum states for any d. Lastly, we also show that the graph isomorphism problem reduces to the tensor isomorphism problem over orthogonal and unitary groups.
Cite as

Zhili Chen, Joshua A. Grochow, Youming Qiao, Gang Tang, and Chuanqi Zhang. On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials III: Actions by Classical Groups. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 31:1-31:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{chen_et_al:LIPIcs.ITCS.2024.31,
  author =	{Chen, Zhili and Grochow, Joshua A. and Qiao, Youming and Tang, Gang and Zhang, Chuanqi},
  title =	{{On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials III: Actions by Classical Groups}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{31:1--31:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.31},
  URN =		{urn:nbn:de:0030-drops-195595},
  doi =		{10.4230/LIPIcs.ITCS.2024.31},
  annote =	{Keywords: complexity class, tensor isomorphism, polynomial isomorphism, group isomorphism, local operations and classical communication}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.4
On the Algebraic Proof Complexity of Tensor Isomorphism

Authors: Nicola Galesi, Joshua A. Grochow, Toniann Pitassi, and Adrian She

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

Abstract
The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz & Grohe, SODA '17). Our main results are an Ω(n) lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank. We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices (which is essentially the same as 2-TI), or deriving BA = I from AB = I. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC+Inv, which allows as derivation rules all substitution instances of the implication AB = I → BA = I. We conjecture that even PC+Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC+Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI.
Cite as

Nicola Galesi, Joshua A. Grochow, Toniann Pitassi, and Adrian She. On the Algebraic Proof Complexity of Tensor Isomorphism. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 4:1-4:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{galesi_et_al:LIPIcs.CCC.2023.4,
  author =	{Galesi, Nicola and Grochow, Joshua A. and Pitassi, Toniann and She, Adrian},
  title =	{{On the Algebraic Proof Complexity of Tensor Isomorphism}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{4:1--4:40},
  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.2023.4},
  URN =		{urn:nbn:de:0030-drops-182748},
  doi =		{10.4230/LIPIcs.CCC.2023.4},
  annote =	{Keywords: Algebraic proof complexity, Tensor Isomorphism, Graph Isomorphism, Polynomial Calculus, Sum-of-Squares, reductions, lower bounds, proof complexity of linear algebra}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.19
Matrix Multiplication via Matrix Groups

Authors: Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans

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

Abstract
In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of groups as a route to obtaining ω = 2, while other families of groups remain potentially viable. In this paper we turn our attention to matrix groups, whose usefulness within this framework was relatively unexplored. We first show that groups of Lie type cannot prove ω = 2 within the group-theoretic approach. This is based on a representation-theoretic argument that identifies the second-smallest dimension of an irreducible representation of a group as a key parameter that determines its viability in this framework. Our proof builds on Gowers' result concerning product-free sets in quasirandom groups. We then give another barrier that rules out certain natural matrix group constructions that make use of subgroups that are far from being self-normalizing. Our barrier results leave open several natural paths to obtain ω = 2 via matrix groups. To explore these routes we propose working in the continuous setting of Lie groups, in which we develop an analogous theory. Obtaining the analogue of ω = 2 in this potentially easier setting is a key challenge that represents an intermediate goal short of actually proving ω = 2. We give two constructions in the continuous setting, each of which evades one of our two barriers.
Cite as

Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans. Matrix Multiplication via Matrix Groups. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{blasiak_et_al:LIPIcs.ITCS.2023.19,
  author =	{Blasiak, Jonah and Cohn, Henry and Grochow, Joshua A. and Pratt, Kevin and Umans, Chris},
  title =	{{Matrix Multiplication via Matrix Groups}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{19:1--19:16},
  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.2023.19},
  URN =		{urn:nbn:de:0030-drops-175226},
  doi =		{10.4230/LIPIcs.ITCS.2023.19},
  annote =	{Keywords: Fast matrix multiplication, representation theory, matrix groups}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.96
Unitary Property Testing Lower Bounds by Polynomials

Authors: Adrian She and Henry Yuen

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

Abstract
We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between QMA and QMA(2), a long standing question in quantum complexity theory.
Cite as

Adrian She and Henry Yuen. Unitary Property Testing Lower Bounds by Polynomials. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 96:1-96:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{she_et_al:LIPIcs.ITCS.2023.96,
  author =	{She, Adrian and Yuen, Henry},
  title =	{{Unitary Property Testing Lower Bounds by Polynomials}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{96:1--96:17},
  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.2023.96},
  URN =		{urn:nbn:de:0030-drops-175995},
  doi =		{10.4230/LIPIcs.ITCS.2023.96},
  annote =	{Keywords: Quantum query complexity, polynomial method, unitary property testing, quantum proofs, invariant theory, quantum recurrence time, entanglement entropy, BQP, QMA, QMA(2)}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.44
If VNP Is Hard, Then so Are Equations for It

Authors: Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, and Anamay Tengse

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee, Kumar, Ramya, Saptharishi and Tengse (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.
Cite as

Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, and Anamay Tengse. If VNP Is Hard, Then so Are Equations for It. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 44:1-44:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{kumar_et_al:LIPIcs.STACS.2022.44,
  author =	{Kumar, Mrinal and Ramya, C. and Saptharishi, Ramprasad and Tengse, Anamay},
  title =	{{If VNP Is Hard, Then so Are Equations for It}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{44:1--44:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.44},
  URN =		{urn:nbn:de:0030-drops-158547},
  doi =		{10.4230/LIPIcs.STACS.2022.44},
  annote =	{Keywords: Computational Complexity, Algebraic Circuits, Algebraic Natural Proofs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.16
On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors

Authors: Joshua A. Grochow and Youming Qiao

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
In this paper we study some classical complexity-theoretic questions regarding Group Isomorphism (GpI). We focus on p-groups (groups of prime power order) with odd p, which are believed to be a bottleneck case for GpI, and work in the model of matrix groups over finite fields. Our main results are as follows. - Although search-to-decision and counting-to-decision reductions have been known for over four decades for Graph Isomorphism (GI), they had remained open for GpI, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from Tensor Isomorphism (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p-groups of class 2 and exponent p. - Despite the widely held belief that p-groups of class 2 and exponent p are the hardest cases of GpI, there was no reduction to these groups from any larger class of groups. Again using methods from Tensor Isomorphism (ibid.), we show the first such reduction, namely from isomorphism testing of p-groups of "small" class and exponent p to those of class two and exponent p. For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI. Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor & Lubotzky, Geom. Dedicata, 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard Correspondence with Tensor Isomorphism-completeness results (Grochow & Qiao, ibid.).
Cite as

Joshua A. Grochow and Youming Qiao. On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 16:1-16:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{grochow_et_al:LIPIcs.CCC.2021.16,
  author =	{Grochow, Joshua A. and Qiao, Youming},
  title =	{{On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{16:1--16:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.16},
  URN =		{urn:nbn:de:0030-drops-142905},
  doi =		{10.4230/LIPIcs.CCC.2021.16},
  annote =	{Keywords: group isomorphism, search-to-decision reduction, counting-to-decision reduction, nilpotent group isomorphism, p-group isomorphism, tensor isomorphism}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.ICALP.2021.5
Algebraic Proof Systems (Invited Talk)

Authors: Toniann Pitassi

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
Given a set of polynomial equations over a field F, how hard is it to prove that they are simultaneously unsolvable? In the last twenty years, algebraic proof systems for refuting such systems of equations have been extensively studied, revealing close connections to both upper bounds (connections between short refutations and efficient approximation algorithms) and lower bounds (connections to fundamental questions in circuit complexity.) The Ideal Proof System (IPS) is a simple yet powerful algebraic proof system, with very close connections to circuit lower bounds: [Joshua A. Grochow and Toniann Pitassi, 2018] proved that lower bounds for IPS imply VNP ≠ VP, and very recently connections in the other direction have been made, showing that circuit lower bounds imply IPS lower bounds [Rahul Santhanam and Iddo Tzameret, 2021; Yaroslav Alekseev et al., 2020]. In this talk I will survey the landscape of algebraic proof systems, focusing on their connections to complexity theory, derandomization, and standard proposional proof complexity. I will discuss the state-of-the-art lower bounds, as well as the relationship between algebraic systems and textbook style propositional proof systems. Finally we end with open problems, and some recent progress towards proving superpolynomial lower bounds for bounded-depth Frege systems with modular gates (a major open problem in propositional proof complexity).
Cite as

Toniann Pitassi. Algebraic Proof Systems (Invited Talk). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, p. 5:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{pitassi:LIPIcs.ICALP.2021.5,
  author =	{Pitassi, Toniann},
  title =	{{Algebraic Proof Systems}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{5:1--5:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.5},
  URN =		{urn:nbn:de:0030-drops-140747},
  doi =		{10.4230/LIPIcs.ICALP.2021.5},
  annote =	{Keywords: complexity theory, proof complexity, algebraic circuits}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.38
Average-Case Algorithms for Testing Isomorphism of Polynomials, Algebras, and Multilinear Forms

Authors: Joshua A. Grochow, Youming Qiao, and Gang Tang

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms f, g ∈ 𝔽_q[x_1, … , x_n], and decides whether f and g are isomorphic in time q^O(n) for most f. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow & Qiao, ITCS 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.
Cite as

Joshua A. Grochow, Youming Qiao, and Gang Tang. Average-Case Algorithms for Testing Isomorphism of Polynomials, Algebras, and Multilinear Forms. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 38:1-38:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{grochow_et_al:LIPIcs.STACS.2021.38,
  author =	{Grochow, Joshua A. and Qiao, Youming and Tang, Gang},
  title =	{{Average-Case Algorithms for Testing Isomorphism of Polynomials, Algebras, and Multilinear Forms}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{38:1--38:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.38},
  URN =		{urn:nbn:de:0030-drops-136836},
  doi =		{10.4230/LIPIcs.STACS.2021.38},
  annote =	{Keywords: polynomial isomorphism, trilinear form equivalence, algebra isomorphism, average-case algorithms, tensor isomorphism complete, symmetric and alternating bilinear maps}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.31
On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness

Authors: Joshua A. Grochow and Youming Qiao

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We study the complexity of isomorphism problems for tensors, groups, and polynomials. These problems have been studied in multivariate cryptography, machine learning, quantum information, and computational group theory. We show that these problems are all polynomial-time equivalent, creating bridges between problems traditionally studied in myriad research areas. This prompts us to define the complexity class TI, namely problems that reduce to the Tensor Isomorphism (TI) problem in polynomial time. Our main technical result is a polynomial-time reduction from d-tensor isomorphism to 3-tensor isomorphism. In the context of quantum information, this result gives multipartite-to-tripartite entanglement transformation procedure, that preserves equivalence under stochastic local operations and classical communication (SLOCC).
Cite as

Joshua A. Grochow and Youming Qiao. On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{grochow_et_al:LIPIcs.ITCS.2021.31,
  author =	{Grochow, Joshua A. and Qiao, Youming},
  title =	{{On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{31:1--31:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.31},
  URN =		{urn:nbn:de:0030-drops-135702},
  doi =		{10.4230/LIPIcs.ITCS.2021.31},
  annote =	{Keywords: complexity class, tensor isomorphism, polynomial isomorphism, group isomorphism, stochastic local operations and classical communication}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.26
Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice

Authors: Peter A. Brooksbank, Yinan Li, Youming Qiao, and James B. Wilson

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
Motivated by testing isomorphism of p-groups, we study the alternating matrix space isometry problem (AltMatSpIso), which asks to decide whether two m-dimensional subspaces of n×n alternating (skew-symmetric if the field is not of characteristic 2) matrices are the same up to a change of basis. Over a finite field 𝔽_p with some prime p≠2, solving AltMatSpIso in time p^O(n+m) is equivalent to testing isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. The latter problem has long been considered a bottleneck case for the group isomorphism problem. Recently, Li and Qiao presented an average-case algorithm for AltMatSpIso in time p^O(n) when n and m are linearly related (FOCS '17). In this paper, we present an average-case algorithm for AltMatSpIso in time p^O(n+m). Besides removing the restriction on the relation between n and m, our algorithm is considerably simpler, and the average-case analysis is stronger. We then implement our algorithm, with suitable modifications, in Magma. Our experiments indicate that it improves significantly over default (brute-force) algorithms for this problem.
Cite as

Peter A. Brooksbank, Yinan Li, Youming Qiao, and James B. Wilson. Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{brooksbank_et_al:LIPIcs.ESA.2020.26,
  author =	{Brooksbank, Peter A. and Li, Yinan and Qiao, Youming and Wilson, James B.},
  title =	{{Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.26},
  URN =		{urn:nbn:de:0030-drops-128920},
  doi =		{10.4230/LIPIcs.ESA.2020.26},
  annote =	{Keywords: Alternating Matrix Spaces, Average-case Algorithm, p-groups of Class 2nd Exponent p, Magma}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.72
Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests

Authors: Janaky Murthy, Vineet Nair, and Chandan Saha

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

Abstract
Equivalence testing for a polynomial family {g_m}_{m ∈ ℕ} over a field 𝔽 is the following problem: Given black-box access to an n-variate polynomial f({𝐱}), where n is the number of variables in g_m for some m ∈ ℕ, check if there exists an A ∈ GL(n,𝔽) such that f({𝐱}) = g_m(A{𝐱}). If yes, then output such an A. The complexity of equivalence testing has been studied for a number of important polynomial families, including the determinant (Det) and the family of iterated matrix multiplication polynomials. Two popular variants of the iterated matrix multiplication polynomial are: IMM_{w,d} (the (1,1) entry of the product of d many w× w symbolic matrices) and Tr-IMM_{w,d} (the trace of the product of d many w× w symbolic matrices). The families - Det, IMM and Tr-IMM - are VBP-complete under p-projections, and so, in this sense, they have the same complexity. But, do they have the same equivalence testing complexity? We show that the answer is "yes" for Det and Tr-IMM (modulo the use of randomness). The above result may appear a bit surprising as the complexity of equivalence testing for IMM and that for Det are quite different over ℚ: a randomized poly-time equivalence testing for IMM over ℚ is known [Neeraj Kayal et al., 2019], whereas [Ankit Garg et al., 2019] showed that equivalence testing for Det over ℚ is integer factoring hard (under randomized reductions and assuming GRH). To our knowledge, the complexity of equivalence testing for Tr-IMM was not known before this work. We show that, despite the syntactic similarity between IMM and Tr-IMM, equivalence testing for Tr-IMM and that for Det are randomized poly-time Turing reducible to each other over any field of characteristic zero or sufficiently large. The result is obtained by connecting the two problems via another well-studied problem in computer algebra, namely the full matrix algebra isomorphism problem (FMAI). In particular, we prove the following: 1) Testing equivalence of polynomials to Tr-IMM_{w,d}, for d ≥ 3 and w ≥ 2, is randomized polynomial-time Turing reducible to testing equivalence of polynomials to Det_w, the determinant of the w × w matrix of formal variables. (Here, d need not be a constant.) 2) FMAI is randomized polynomial-time Turing reducible to equivalence testing (in fact, to tensor isomorphism testing) for the family of matrix multiplication tensors {Tr-IMM_{w,3}}_{w ∈ ℕ}. These results, in conjunction with the randomized poly-time reduction (shown in [Ankit Garg et al., 2019]) from determinant equivalence testing to FMAI, imply that the four problems - FMAI, equivalence testing for Tr-IMM and for Det, and the 3-tensor isomorphism problem for the family of matrix multiplication tensors - are randomized poly-time equivalent under Turing reductions.
Cite as

Janaky Murthy, Vineet Nair, and Chandan Saha. Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{murthy_et_al:LIPIcs.MFCS.2020.72,
  author =	{Murthy, Janaky and Nair, Vineet and Saha, Chandan},
  title =	{{Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{72:1--72:16},
  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.2020.72},
  URN =		{urn:nbn:de:0030-drops-127419},
  doi =		{10.4230/LIPIcs.MFCS.2020.72},
  annote =	{Keywords: equivalence testing, determinant, trace of the matrix product, full-matrix algebra isomorphism}
}
  • Refine by Author
  • 9 Grochow, Joshua A.
  • 6 Qiao, Youming
  • 2 Pitassi, Toniann
  • 2 She, Adrian
  • 2 Tang, Gang
  • Show More...

  • Refine by Classification
  • 7 Theory of computation → Algebraic complexity theory
  • 4 Theory of computation → Proof complexity
  • 3 Computing methodologies → Algebraic algorithms
  • 2 Computing methodologies → Combinatorial algorithms
  • 2 Theory of computation → Circuit complexity
  • Show More...

  • Refine by Keyword
  • 3 group isomorphism
  • 3 polynomial isomorphism
  • 3 tensor isomorphism
  • 2 Graph Isomorphism
  • 2 algebraic circuits
  • Show More...

  • Refine by Type
  • 19 document

  • Refine by Publication Year
  • 5 2024
  • 4 2021
  • 3 2023
  • 2 2018
  • 2 2020
  • Show More...

© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact