4 Search Results for "Anderson, Matthew"


Document
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
Media Exposition
An Interactive Tool for Experimenting with Bounded-Degree Plane Geometric Spanners (Media Exposition)

Authors: Fred Anderson, Anirban Ghosh, Matthew Graham, Lucas Mougeot, and David Wisnosky

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


Abstract
The construction of bounded-degree plane geometric spanners has been a focus of interest in the field of geometric spanners for a long time. To date, several algorithms have been designed with various trade-offs in degree and stretch factor. Using JSXGraph, a state-of-the-art JavaScript library for geometry, we have implemented seven of these sophisticated algorithms so that they can be used for further research and teaching computational geometry. We believe that our interactive tool can be used by researchers from related fields to understand and apply the algorithms in their research. Our tool can be run in any modern browser. The tool will be permanently maintained by the second author at https://ghoshanirban.github.io/bounded-degree-plane-spanners/index.html

Cite as

Fred Anderson, Anirban Ghosh, Matthew Graham, Lucas Mougeot, and David Wisnosky. An Interactive Tool for Experimenting with Bounded-Degree Plane Geometric Spanners (Media Exposition). In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 61:1-61:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{anderson_et_al:LIPIcs.SoCG.2021.61,
  author =	{Anderson, Fred and Ghosh, Anirban and Graham, Matthew and Mougeot, Lucas and Wisnosky, David},
  title =	{{An Interactive Tool for Experimenting with Bounded-Degree Plane Geometric Spanners}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{61:1--61:4},
  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.2021.61},
  URN =		{urn:nbn:de:0030-drops-138607},
  doi =		{10.4230/LIPIcs.SoCG.2021.61},
  annote =	{Keywords: graph approximation, Delaunay triangulations, geometric spanners, plane spanners, bounded-degree spanners}
}
Document
Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs

Authors: Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, and Ben Lee Volk

Published in: LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)


Abstract
Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs). In this work, we give an exponential lower bound of exp(n/k^{O(k)}) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2^{~O(n^{1-1/2^{k-1}})} and needs white box access only to know the order in which the variables appear in the ABP.

Cite as

Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, and Ben Lee Volk. Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 30:1-30:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{anderson_et_al:LIPIcs.CCC.2016.30,
  author =	{Anderson, Matthew and Forbes, Michael A. and Saptharishi, Ramprasad and Shpilka, Amir and Volk, Ben Lee},
  title =	{{Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs}},
  booktitle =	{31st Conference on Computational Complexity (CCC 2016)},
  pages =	{30:1--30:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-008-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{50},
  editor =	{Raz, Ran},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.30},
  URN =		{urn:nbn:de:0030-drops-58255},
  doi =		{10.4230/LIPIcs.CCC.2016.30},
  annote =	{Keywords: Algebraic Complexity, Lower Bounds, Derandomization, Polynomial Identity Testing}
}
Document
On Symmetric Circuits and Fixed-Point Logics

Authors: Matthew Anderson and Anuj Dawar

Published in: LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)


Abstract
We study properties of relational structures such as graphs that are decided by families of Boolean circuits. Circuits that decide such properties are necessarily invariant to permutations of the elements of the input structures. We focus on families of circuits that are symmetric, i.e., circuits whose invariance is witnessed by automorphisms of the circuit induced by the permutation of the input structure. We show that the expressive power of such families is closely tied to definability in logic. In particular, we show that the queries defined on structures by uniform families of symmetric Boolean circuits with majority gates are exactly those definable in fixed-point logic with counting. This shows that inexpressibility results in the latter logic lead to lower bounds against polynomial-size families of symmetric circuits.

Cite as

Matthew Anderson and Anuj Dawar. On Symmetric Circuits and Fixed-Point Logics. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 41-52, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


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@InProceedings{anderson_et_al:LIPIcs.STACS.2014.41,
  author =	{Anderson, Matthew and Dawar, Anuj},
  title =	{{On Symmetric Circuits and Fixed-Point Logics}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{41--52},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Mayr, Ernst W. and Portier, Natacha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.41},
  URN =		{urn:nbn:de:0030-drops-44455},
  doi =		{10.4230/LIPIcs.STACS.2014.41},
  annote =	{Keywords: symmetric circuit, fixed-point logic, majority, counting, uniformity}
}
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