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
nonsingular matrix completion ($\NSMC$) under a lowindividualdegree 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}, hardnessrandomness 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 blackbox 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 NonSingular Matrix Completion}},
booktitle = {27th International Symposium on Theoretical Aspects of Computer Science},
pages = {465476},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897163},
ISSN = {18688969},
year = {2010},
volume = {5},
editor = {JeanYves Marion and Thomas Schwentick},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2010/2477},
URN = {urn:nbn:de:0030drops24770},
doi = {10.4230/LIPIcs.STACS.2010.2477},
annote = {Keywords: Computational complexity, arithmetic circuits, hardnessrandomness tradeoffs, identity testing, determinant versus permanent}
}
Keywords: 

Computational complexity, arithmetic circuits, hardnessrandomness tradeoffs, identity testing, determinant versus permanent 
Collection: 

27th International Symposium on Theoretical Aspects of Computer Science 
Issue Date: 

2010 
Date of publication: 

09.03.2010 