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Electronic Colloquium on Computational Complexity

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Reports tagged with VP:
TR13-091 | 17th June 2013
Neeraj Kayal, Chandan Saha, Ramprasad Saptharishi

A super-polynomial lower bound for regular arithmetic formulas.

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 >>>

TR14-163 | 29th November 2014
Arnaud Durand, Meena Mahajan, Guillaume Malod, Nicolas de Rugy-Altherre, Nitin Saurabh

Homomorphism polynomials complete for VP

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 >>>

TR16-038 | 15th March 2016
Meena Mahajan, Nitin Saurabh

Some Complete and Intermediate Polynomials in Algebraic Complexity Theory

Revisions: 2

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 ... more >>>

TR17-153 | 9th October 2017
Pranjal Dutta, Nitin Saxena, Amit Sinhababu

Discovering the roots: Uniform closure results for algebraic classes under factoring

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 >>>

TR18-124 | 6th July 2018
Amir Yehudayoff

Separating Monotone VP and VNP

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.

more >>>

TR20-039 | 25th March 2020
Pranjal Dutta, Nitin Saxena, Thomas Thierauf

Lower bounds on the sum of 25th-powers of univariates lead to complete derandomization of PIT

We consider the univariate polynomial $f_d:=(x+1)^d$ when represented as a sum of constant-powers of univariate polynomials. We define a natural measure for the model, the support-union, and conjecture that it is $\Omega(d)$ for $f_d$.

We show a stunning connection of the conjecture to the two main problems in algebraic ... more >>>

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