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Combinatorica

, Volume 38, Issue 5, pp 1205–1238 | Cite as

Black-Box Identity Testing of Depth-4 Multilinear Circuits

  • Shubhangi Saraf
  • Ilya Volkovich
Original Paper
  • 12 Downloads

Abstract

We study the problem of identity testing for multilinear ΣΠΣΠ(k) circuits, i.e., multilinear depth-4 circuits with fan-in k at the top + gate. We give the first polynomial-time deterministic identity testing algorithm for such circuits when k=O(1). Our results also hold in the black-box setting.

The running time of our algorithm is \({\left( {ns} \right)^{{\text{O}}\left( {{k^3}} \right)}}\), where n is the number of variables, s is the size of the circuit and k is the fan-in of the top gate. The importance of this model arises from [11], where it was shown that derandomizing black-box polynomial identity testing for general depth-4 circuits implies a derandomization of polynomial identity testing (PIT) for general arithmetic circuits. Prior to our work, the best PIT algorithm for multilinear ΣΠΣΠ(k) circuits [31] ran in quasi-polynomial-time, with the running time being \({n^{{\rm O}\left( {{k^6}\log \left( k \right){{\log }^2}s} \right)}}\).

We obtain our results by showing a strong structural result for multilinear ΣΠΣΠ(k) circuits that compute the zero polynomial. We show that under some mild technical conditions, any gate of such a circuit must compute a sparse polynomial. We then show how to combine the structure theorem with a result by Klivans and Spielman [33], on the identity testing for sparse polynomials, to yield the full result.

Mathematics Subject Classification (2000)

68Q25 12Y05 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics & Department of Computer ScienceRutgers UniversityPiscatawayUSA
  2. 2.Department of EECS, CSE DivisionUniversity of MichiganAnn ArborUSA

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