4 Search Results for "Spielman, Daniel A."

Document
DOI: 10.4230/LIPIcs.ESA.2023.29
Effective Resistances in Non-Expander Graphs

Authors: Dongrun Cai, Xue Chen, and Pan Peng

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
Effective resistances are ubiquitous in graph algorithms and network analysis. For an undirected graph G, its effective resistance R_G(s,t) between two vertices s and t is defined as the equivalent resistance between s and t if G is thought of as an electrical network with unit resistance on each edge. If we use L_G to denote the Laplacian matrix of G and L_G^† to denote its pseudo-inverse, we have R_G(s,t) = (𝟏_s-𝟏_t)^⊤ L^† (𝟏_s-𝟏_t) such that classical Laplacian solvers [Daniel A. Spielman and Shang{-}Hua Teng, 2014] provide almost-linear time algorithms to approximate R_G(s,t). In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair s and t. We consider the classical adjacency list model [Ron, 2019] for local algorithms. While recent works [Andoni et al., 2018; Peng et al., 2021; Li and Sachdeva, 2023] have provided sublinear time algorithms for expander graphs, we prove several lower bounds for general graphs of n vertices and m edges: 1) It needs Ω(n) queries to obtain 1.01-approximations of the effective resistance of an adjacent pair s and t, even for graphs of degree at most 3 except s and t. 2) For graphs of degree at most d and any parameter 𝓁, it needs Ω(m/𝓁) queries to obtain c ⋅ min{d,𝓁}-approximations where c > 0 is a universal constant. Moreover, we supplement the first lower bound by providing a sublinear time (1+ε)-approximation algorithm for graphs of degree 2 except the pair s and t. One of our technical ingredients is to bound the expansion of a graph in terms of the smallest non-trivial eigenvalue of its Laplacian matrix after removing edges. We discover a new lower bound on the eigenvalues of perturbed graphs (resp. perturbed matrices) by incorporating the effective resistance of the removed edge (resp. the leverage scores of the removed rows), which may be of independent interest.
Cite as

Dongrun Cai, Xue Chen, and Pan Peng. Effective Resistances in Non-Expander Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{cai_et_al:LIPIcs.ESA.2023.29,
  author =	{Cai, Dongrun and Chen, Xue and Peng, Pan},
  title =	{{Effective Resistances in Non-Expander Graphs}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{29:1--29:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.29},
  URN =		{urn:nbn:de:0030-drops-186823},
  doi =		{10.4230/LIPIcs.ESA.2023.29},
  annote =	{Keywords: Sublinear Time Algorithm, Effective Resistance, Leverage Scores, Matrix Perturbation}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.30
A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph

Authors: Dmitriy Kunisky and Xifan Yu

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

Abstract
We prove that the degree 4 sum-of-squares (SOS) relaxation of the clique number of the Paley graph on a prime number p of vertices has value at least Ω(p^{1/3}). This is in contrast to the widely believed conjecture that the actual clique number of the Paley graph is O(polylog(p)). Our result may be viewed as a derandomization of that of Deshpande and Montanari (2015), who showed the same lower bound (up to polylog(p) terms) with high probability for the Erdős-Rényi random graph on p vertices, whose clique number is with high probability O(log(p)). We also show that our lower bound is optimal for the Feige-Krauthgamer construction of pseudomoments, derandomizing an argument of Kelner. Finally, we present numerical experiments indicating that the value of the degree 4 SOS relaxation of the Paley graph may scale as O(p^{1/2 - ε}) for some ε > 0, and give a matrix norm calculation indicating that the pseudocalibration construction for SOS lower bounds for random graphs will not immediately transfer to the Paley graph. Taken together, our results suggest that degree 4 SOS may break the "√p barrier" for upper bounds on the clique number of Paley graphs, but prove that it can at best improve the exponent from 1/2 to 1/3.
Cite as

Dmitriy Kunisky and Xifan Yu. A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 30:1-30:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{kunisky_et_al:LIPIcs.CCC.2023.30,
  author =	{Kunisky, Dmitriy and Yu, Xifan},
  title =	{{A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{30:1--30:25},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.30},
  URN =		{urn:nbn:de:0030-drops-183008},
  doi =		{10.4230/LIPIcs.CCC.2023.30},
  annote =	{Keywords: convex optimization, sum of squares, Paley graph, derandomization}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.40
Hardness Results for Weaver’s Discrepancy Problem

Authors: Daniel A. Spielman and Peng Zhang

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract
Marcus, Spielman and Srivastava (Annals of Mathematics 2014) solved the Kadison-Singer Problem by proving a strong form of Weaver’s conjecture: they showed that for all α > 0 and all lists of vectors of norm at most √α whose outer products sum to the identity, there exists a signed sum of those outer products with operator norm at most √{8α} + 2α. We prove that it is NP-hard to distinguish such a list of vectors for which there is a signed sum that equals the zero matrix from those in which every signed sum has operator norm at least η √α, for some absolute constant η > 0. Thus, it is NP-hard to construct a signing that is a constant factor better than that guaranteed to exist. For α = 1/4, we prove that it is NP-hard to distinguish whether there is a signed sum that equals the zero matrix from the case in which every signed sum has operator norm at least 1/4.
Cite as

Daniel A. Spielman and Peng Zhang. Hardness Results for Weaver’s Discrepancy Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{spielman_et_al:LIPIcs.APPROX/RANDOM.2022.40,
  author =	{Spielman, Daniel A. and Zhang, Peng},
  title =	{{Hardness Results for Weaver’s Discrepancy Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{40:1--40:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.40},
  URN =		{urn:nbn:de:0030-drops-171628},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.40},
  annote =	{Keywords: Discrepancy Problem, Kadison-Singer Problem, Hardness of Approximation}
}
Document
DOI: 10.4230/DagSemProc.09061.3
A Nearly-Linear Time Algorithm for Approximately Solving Linear Systems in a Symmetric M-Matrix

Authors: Samuel I. Daitch and Daniel A. Spielman

Published in: Dagstuhl Seminar Proceedings, Volume 9061, Combinatorial Scientific Computing (2009)

Abstract
We present an algorithm for solving a linear system in a symmetric M-matrix. In particular, for $n times n$ symmetric M-matrix $M$, we show how to find a diagonal matrix $D$ such that $DMD$ is diagonally-dominant. To compute $D$, the algorithm must solve $O{log n}$ linear systems in diagonally-dominant matrices. If we solve these diagonally-dominant systems approximately using the Spielman-Teng nearly-linear time solver, then we obtain an algorithm for approximately solving linear systems in symmetric M-matrices, for which the expected running time is also nearly-linear.
Cite as

Samuel I. Daitch and Daniel A. Spielman. A Nearly-Linear Time Algorithm for Approximately Solving Linear Systems in a Symmetric M-Matrix. In Combinatorial Scientific Computing. Dagstuhl Seminar Proceedings, Volume 9061, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

Copy BibTex To Clipboard

@InProceedings{daitch_et_al:DagSemProc.09061.3,
  author =	{Daitch, Samuel I. and Spielman, Daniel A.},
  title =	{{A Nearly-Linear Time Algorithm for Approximately Solving Linear Systems in a Symmetric M-Matrix}},
  booktitle =	{Combinatorial Scientific Computing},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{9061},
  editor =	{Uwe Naumann and Olaf Schenk and Horst D. Simon and Sivan Toledo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09061.3},
  URN =		{urn:nbn:de:0030-drops-20803},
  doi =		{10.4230/DagSemProc.09061.3},
  annote =	{Keywords: M-matrix, diagonally-dominant matrix, linear system solver, iterative algorithm, randomized algorithm, network flow, gain graph}
}
  • Refine by Author
  • 2 Spielman, Daniel A.
  • 1 Cai, Dongrun
  • 1 Chen, Xue
  • 1 Daitch, Samuel I.
  • 1 Kunisky, Dmitriy
  • Show More...

  • Refine by Classification
  • 1 Mathematics of computing → Combinatorial optimization
  • 1 Theory of computation → Design and analysis of algorithms
  • 1 Theory of computation → Problems, reductions and completeness
  • 1 Theory of computation → Pseudorandomness and derandomization
  • 1 Theory of computation → Semidefinite programming
  • Show More...

  • Refine by Keyword
  • 1 Discrepancy Problem
  • 1 Effective Resistance
  • 1 Hardness of Approximation
  • 1 Kadison-Singer Problem
  • 1 Leverage Scores
  • Show More...

  • Refine by Type
  • 4 document

  • Refine by Publication Year
  • 2 2023
  • 1 2009
  • 1 2022

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact