Neeraj Kayal, Chandan Saha, Ramprasad Saptharishi

We consider arithmetic formulas consisting of alternating layers of addition $(+)$ and multiplication $(\times)$ gates such that the fanin of all the gates in any fixed layer is the same. Such a formula $\Phi$ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree ... more >>>

Arnaud Durand, Meena Mahajan, Guillaume Malod, Nicolas de Rugy-Altherre, Nitin Saurabh

The VP versus VNP question, introduced by Valiant, is probably the most important open question in algebraic complexity theory. Thanks to completeness results, a variant of this question, VBP versus VNP, can be succinctly restated as asking whether the permanent of a generic matrix can be written as a determinant ... more >>>

Meena Mahajan, Nitin Saurabh

We provide a list of new natural VNP-intermediate polynomial

families, based on basic (combinatorial) NP-complete problems that

are complete under \emph{parsimonious} reductions. Over finite

fields, these families are in VNP, and under the plausible

hypothesis $\text{Mod}_pP \not\subseteq P/\text{poly}$, are neither VNP-hard (even under

oracle-circuit reductions) nor in VP. Prior to ...
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Pranjal Dutta, Nitin Saxena, Amit Sinhababu

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the ... more >>>

Amir Yehudayoff

This work is about the monotone versions of the algebraic complexity classes VP and VNP. The main result is that monotone VNP is strictly stronger than monotone VP.