Free access
Proceedings
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

Near-optimal Bootstrapping of Hitting Sets for Algebraic Circuits

Abstract

The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel states that any nonzero polynomial f(xi, …, xn) of degree at most s will evaluate to a nonzero value at some point on a grid with |S| > s. Thus, there is a deterministic polynomial identity test (PIT) for all degrees size-s algebraic circuits in n variables that runs in time poly(s) · (s + 1)n. In a surprising recent result, Agrawal, Ghosh and Saxena (STOC 2018) showed any deterministic blackbox PIT algorithm for degree-s, size-s, n-variate circuits with running time as bad as (sn0.5−δ) Huge(n), where δ > 0 and Huge(n) is an arbitrary function, can be used to construct blackbox PIT algorithms for degree-s size s circuits with running time sexp(exp(O(log* s))). Agrawal et al. asked if a similar conclusion followed if their hypothesis was weakened to having deterministic PIT with running time so(n) · Huge(n). In this paper, we answer their question in the affirmative. We show that, given a deterministic blackbox PIT that runs in time so(n) · Huge(n) for all degree-s size-s algebraic circuits over n variables, we can obtain a deterministic blackbox PIT that runs in time sexp(exp(O(log* s))) for all degree-s size-s algebraic circuits over n variables. In other words, any blackbox PIT with just a slightly nontrivial exponent of s compared to the trivial sO(n) test can be used to give a nearly polynomial time blackbox PIT algorithm.

Formats available

You can view the full content in the following formats:

Information & Authors

Information

Published In

cover image Proceedings
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 639 - 646
Editor: Timothy M. Chan, University of Illinois at Urbana-Champaign, USA
ISBN (Online): 978-1-61197-548-2

History

Published online: 2 January 2019

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

Figures

Tables

Media

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media