Some intersection theorems for ordered sets and graphs
Journal of Combinatorial Theory Series A
Matching is as easy as matrix inversion
Combinatorica
A deterministic algorithm for sparse multivariate polynomial interpolation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Algebraic methods for interactive proof systems
Journal of the ACM (JACM)
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Randomness efficient identity testing of multivariate polynomials
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Deterministic identity testing for multivariate polynomials
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Testing polynomials which are easy to compute (Extended Abstract)
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Combinatorics, Probability and Computing
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
Polynomial Identity Testing for Depth 3 Circuits
Computational Complexity
Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits
SIAM Journal on Computing
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
IP=PSPACE (interactive proof=polynomial space)
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
Diagonal Circuit Identity Testing and Lower Bounds
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Arithmetic Circuits: A Chasm at Depth Four
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Multi-linear formulas for permanent and determinant are of super-polynomial size
Journal of the ACM (JACM)
An Almost Optimal Rank Bound for Depth-3 Identities
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
Improved Polynomial Identity Testing for Read-Once Formulas
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits
SIAM Journal on Computing
The ideal membership problem and polynomial identity testing
Information and Computation
Blackbox Polynomial Identity Testing for Depth 3 Circuits
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in
Proceedings of the forty-second ACM symposium on Theory of computing
On the relation between polynomial identity testing and finding variable disjoint factors
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits
SIAM Journal on Computing
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter
Proceedings of the forty-third annual ACM symposium on Theory of computing
Recent results on polynomial identity testing
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
Algebraic independence and blackbox identity testing
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Reconstruction of depth-4 multilinear circuits with top fan-in 2
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Algebraic independence and blackbox identity testing
Information and Computation
On Enumerating Monomials and Other Combinatorial Structures by Polynomial Interpolation
Theory of Computing Systems
Hi-index | 0.00 |
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. Our results also hold in the black-box setting. The running time of our algorithm is (ns)O(k3), 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 [3], 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 [13] ran in quasi-polynomial-time, with the running time being nO(k6 log(k) log2 s ). 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 [17], on the identity testing for sparse polynomials, to yield the full result.