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URN: urn:nbn:de:0030-drops-50548
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Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

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Abstract

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models. For depth-3 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 2*delta/3))). This implies a lower bound of exp(~Omega(n^(1/2))) for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 4*delta/3)). This implies a lower bound of exp(~Omega(n^(1/4))) for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of +,* gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp(n^(1-delta)), for regular depth-d multilinear formulas of size exp(n^delta), where delta = O(1/sqrt(5)^d)). This result implies a lower bound of roughly exp(~Omega(n^(1/sqrt(5)^d))) for such formulas. We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known. Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs).

BibTeX - Entry

```@InProceedings{oliveira_et_al:LIPIcs:2015:5054,
author =	{Rafael Oliveira and Amir Shpilka and Ben Lee Volk},
title =	{{Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas}},
booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
pages =	{304--322},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-81-1},
ISSN =	{1868-8969},
year =	{2015},
volume =	{33},
editor =	{David Zuckerman},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},