Paper Info
Title
Quasi-polynomial hitting-set for set-depth-Δ formulas
Abstract
We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition X1⊔⋅⋅⋅⊔ Xd of the variable indices [n] that the top product layer respects, i.e. C(term{x})=∑i=1k ∏j=1d fi,j(term{x}Xj), where fi,j is a sparse polynomial in F[term{x}Xj]. Extending this definition to any depth - we call a depth-D formula C (consisting of alternating layers of Σ and Π gates, with a Σ-gate on top) a set-depth-D formula if every Π-layer in C respects a (unknown) partition on the variables; if D is even then the product gates of the bottom-most Π-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-D formulas (over any field) with running time polynomial in exp((D2log s) Δ - 1), where s is the size bound on the input set-depth-D formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of D=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995) and recently by Forbes & Shpilka (STOC 2012 & ECCC TR12-115). Our work settles this question, not only for depth-3 but, up to depth εlog s / log log s, for a fixed constant ε Hadamard algebra, after applying a 'shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-D formulas.
Year
DOI
Venue
2013
10.1145/2488608.2488649
Electronic Colloquium on Computational Complexity (ECCC)
Keywords
Field
DocType
quasi-polynomial hitting-set,depth-d formula,set-depth-d formula,depth-4 formula,sub-exponential time hitting-set generator,log log,constant-depth formula,input set-depth-d formula,sparse polynomial,quasi-polynomial time blackbox polynomial,latter formula
Log-log plot,Discrete mathematics,Polynomial identity testing,Combinatorics,Open problem,Polynomial,Quasi-polynomial,e,Monomial,Partition (number theory),Mathematics
Conference
Volume
Citations 
PageRank 
abs/1209.2333
22
0.67
References 
Authors
27
3
Name
Order
Citations
PageRank
Manindra Agrawal158145.56
Chandan Saha222716.91
Nitin Saxena328026.72