Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion

Kabanets and Impagliazzo [KI04] show how to decide the circuit polynomial identity testing problem (CPIT) in deterministic subexponential time, assuming hardness of some explicit multilinear polynomial family {fm}m≥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 determinantal complexity dc(fm) of fm. 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 × m permanent is mω(log m) . In this paper it is shown, for an appropriate notion of explicitness, that the existence of an explicit multilinear polynomial family {fm}m≥1 with dc(fm) = mω(lo...

• Type: Conference
• Year: 2010
• Where: STACS
• Authors: Maurice Jansen
• Keywords: Explicit Multilinear Polynomial, Multilinear Nsmc, Multilinear Polynomial Family, STACS 2010, Theoretical Computer Science