Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion

In (Kabanets, Impagliazzo, 2004) it is shown how to decide the circuit polynomial identity testing problem (CPIT) in deterministic subexponential time, assuming hardness of some explicit multilinear polynomial family for arithmetical circuits. In this paper, a special case of CPIT is considered, namely low-degree non-singular matrix completion (NSMC). For this subclass of problems it is shown how to obtain the same deterministic time bound, using a weaker assumption in terms of determinantal complexity. 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-by-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 with determinantal complexity m^{\omega(\log m)}$ is equivalent to the existence of an efficiently computable generator $G_n$ 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.

💡 Research Summary

The paper revisits the deterministic sub‑exponential‑time algorithm for circuit polynomial identity testing (CPIT) originally obtained by Kabanets and Impagliazzo (2004). Their result required a strong hardness assumption: the existence of an explicit multilinear polynomial family that is hard for arithmetic circuits. The authors focus on a restricted CPIT problem, namely low‑degree non‑singular matrix completion (NSMC). In NSMC one is given an m‑by‑m matrix M(x) whose entries are polynomials over n variables, and the task is to find an assignment to the variables that makes det(M(x)) non‑zero.

The first contribution is to replace the “circuit‑hardness” assumption with a weaker, more concrete one expressed in terms of determinantal complexity (the minimal size of a matrix whose determinant equals the polynomial). They show that if there exists an explicit multilinear family {fₙ} whose determinantal complexity satisfies dc(fₙ) ≥ n^{ω(log n)}, then NSMC can be solved deterministically in time 2^{Õ(√n)} – the same bound as in the original CPIT result. The proof adapts the Kabanets‑Impagliazzo framework to the NSMC setting, exploiting the fact that a high determinantal complexity forces any small‑seed generator to hit a non‑zero determinant.

A second, complementary result establishes a hardness‑randomness trade‑off in the opposite direction. The authors construct a black‑box generator Gₙ with seed length s = O(n^{1/√log n}) that works for multilinear NSMC (the case where det(M(x)) itself is multilinear and M has polynomial size). They prove that the existence of such a generator is equivalent to the existence of an explicit multilinear family with determinantal complexity n^{ω(log n)}. In other words, a short‑seed generator for multilinear NSMC exists if and only if there are explicit polynomials whose determinantal complexity grows super‑polynomially in the logarithm of the number of variables.

The paper then connects these findings to the long‑standing VP versus VNP problem. Valiant showed that separating VP from VNP would follow from proving that the m‑by‑m permanent has determinantal complexity m^{ω(log m)}. By the equivalence above, such a lower bound is tantamount to the existence of an efficiently computable generator for multilinear NSMC with the aforementioned seed length. Thus, progress on constructing generators for NSMC directly translates into progress on lower bounds for the permanent, and vice‑versa.

Technical highlights include:

• Definition of multilinear NSMC – the matrix size is poly(n) and its determinant is multilinear, which allows the authors to apply dimension‑reduction and sampling techniques more finely than in the general CPIT setting.
• Construction of Gₙ – the generator enumerates all seeds of length s, each producing a full assignment to the n variables. Because s = O(n^{1/√log n}), the total enumeration runs in 2^{Õ(√n)} time, matching the sub‑exponential bound.
• Bidirectional equivalence proof – from high determinantal complexity to the existence of Gₙ, the authors show that any hard polynomial forces a hitting set of the required size; conversely, a hitting set of that size yields a low‑dimensional determinantal representation, contradicting the hardness unless the complexity is indeed high.
• Implications for VP/VNP – the equivalence reframes the permanent lower‑bound question as a generator‑construction problem, offering a new avenue for attacking Valiant’s conjecture.

The authors acknowledge limitations: the seed length O(n^{1/√log n}) is still far from polylogarithmic, and the results apply only to the multilinear NSMC subclass rather than full CPIT. Nevertheless, the work demonstrates that deterministic sub‑exponential algorithms for important algebraic problems can be achieved under substantially weaker, more quantifiable assumptions, and it forges a concrete bridge between algorithmic derandomization and algebraic circuit lower bounds.