When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2010.2477
URN: urn:nbn:de:0030-drops-24770
URL: http://drops.dagstuhl.de/opus/volltexte/2010/2477/
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### Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion

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### Abstract

Kabanets and Impagliazzo \cite{KaIm04} show how to decide the circuit polynomial identity testing problem (CPIT) in deterministic subexponential time, assuming hardness of some explicit multilinear polynomial family $\{f_m\}_{m \geq 1}$ for arithmetic circuits. In this paper, a special case of CPIT is considered, namely non-singular matrix completion ($\NSMC$) under a low-individual-degree promise. For this subclass of problems it is shown how to obtain the same deterministic time bound, using a weaker assumption in terms of the {\em determinantal complexity} $\dcomp(f_m)$ of $f_m$. Building on work by Agrawal \cite{Agr05}, hardness-randomness tradeoffs will also be shown in the converse direction, in an effort to make progress on Valiant's $\VP$ versus $\VNP$ problem. To separate $\VP$ and $\VNP$, it is known to be sufficient to prove that the determinantal complexity of the $m\times m$ permanent is $m^{\omega(\log m)}$. In this paper it is shown, for an appropriate notion of explicitness, that the existence of an explicit multilinear polynomial family $\{f_m\}_{m \geq 1}$ with $\dcomp(f_m) = m^{\omega(\log m)}$ is equivalent to the existence of an efficiently computable {\em generator} $\{G_n\}_{n\geq 1}$ {\em for} multilinear $\NSMC$ with seed length $O(n^{1/\sqrt{\log n}})$. The latter is a combinatorial object that provides an efficient deterministic black-box algorithm for $\NSMC$. Multilinear $\NSMC$'' indicates that $G_n$ only has to work for matrices $M(x)$ of $poly(n)$ size in $n$ variables, for which $\det(M(x))$ is a multilinear polynomial.

### BibTeX - Entry

@InProceedings{jansen:LIPIcs:2010:2477,
author =	{Maurice Jansen},
title =	{{Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion}},
booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
pages =	{465--476},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-16-3},
ISSN =	{1868-8969},
year =	{2010},
volume =	{5},
editor =	{Jean-Yves Marion and Thomas Schwentick},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2477},
URN =		{urn:nbn:de:0030-drops-24770},
doi =		{10.4230/LIPIcs.STACS.2010.2477},
annote =	{Keywords: Computational complexity, arithmetic circuits, hardness-randomness tradeoffs, identity testing,   determinant versus permanent}
}


 Keywords: Computational complexity, arithmetic circuits, hardness-randomness tradeoffs, identity testing, determinant versus permanent Seminar: 27th International Symposium on Theoretical Aspects of Computer Science Issue Date: 2010 Date of publication: 09.03.2010

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