Document

APPROX

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

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.

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)

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@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.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

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

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.

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)

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@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.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} }

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