Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion

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.