10.4230/LIPICS.STACS.2010.2477
Jansen, Maurice
Maurice
Jansen
Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2010
Computational complexity
arithmetic circuits
hardness-randomness tradeoffs
identity testing
determinant versus permanent
Marion, Jean-Yves
Jean-Yves
Marion
Schwentick, Thomas
Thomas
Schwentick
2010
2010-03-09
2010-03-09
2010-03-09
en
urn:nbn:de:0030-drops-24770
10.4230/LIPIcs.STACS.2010
978-3-939897-16-3
1868-8969
10.4230/LIPIcs.STACS.2010
LIPIcs, Volume 5, STACS 2010
27th International Symposium on Theoretical Aspects of Computer Science
2013
5
41
465
476
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Marion, Jean-Yves
Jean-Yves
Marion
Schwentick, Thomas
Thomas
Schwentick
1868-8969
Leibniz International Proceedings in Informatics (LIPIcs)
2010
5
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
12 pages
333105 bytes
application/pdf
Creative Commons Attribution-NoDerivs 3.0 Unported license
info:eu-repo/semantics/openAccess
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.
LIPIcs, Vol. 5, 27th International Symposium on Theoretical Aspects of Computer Science, pages 465-476