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Electronic Colloquium on Computational Complexity

Under the auspices of the Computational Complexity Foundation (CCF)

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Reports tagged with Border Complexity:
TR96-058 | 25th November 1996
Dima Grigoriev, Marek Karpinski

Randomized $\mathbf{\Omega (n^2)}$ Lower Bound for Knapsack

We prove $\Omega (n^2)$ complexity \emph{lower bound} for the
general model of \emph{randomized computation trees} solving
the \emph{Knapsack Problem}, and more generally \emph{Restricted
Integer Programming}. This is the \emph{first nontrivial} lower
bound proven for this model of computation. The method of the ... more >>>

TR17-034 | 21st February 2017
Karl Bringmann, Christian Ikenmeyer, Jeroen Zuiddam

On algebraic branching programs of small width

Revisions: 1

In 1979 Valiant showed that the complexity class VP_e of families with polynomially bounded formula size is contained in the class VP_s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes we study the topological closure VP_e-bar, i.e. the ... more >>>

TR18-068 | 8th April 2018
Mrinal Kumar

On top fan-in vs formal degree for depth-3 arithmetic circuits

Revisions: 1

We show that over the field of complex numbers, every homogeneous polynomial of degree $d$ can be approximated (in the border complexity sense) by a depth-$3$ arithmetic circuit of top fan-in at most $d+1$. This is quite surprising since there exist homogeneous polynomials $P$ on $n$ variables of degree $2$, ... more >>>

TR19-065 | 1st May 2019
Mrinal Kumar, Ramprasad Saptharishi, Noam Solomon

Derandomization from Algebraic Hardness: Treading the Borders

Revisions: 2

A hitting-set generator (HSG) is a polynomial map $Gen:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $Q$ of small enough circuit size and degree, if $Q$ is non-zero, then $Q\circ Gen$ is non-zero. In this paper, we give a new construction of such a HSG assuming that we have ... more >>>

TR20-031 | 10th March 2020
Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, Nitin Saurabh

Algebraic Branching Programs, Border Complexity, and Tangent Spaces

Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices. This means that the set of noncommutative polynomials with ABP width complexity at most $k$ is Zariski-closed, an important property in ... more >>>

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