Perceptrons: expanded edition
Journal of the ACM (JACM)
Fixed-parameter tractability and completeness II: on completeness for W[1]
Theoretical Computer Science
Randomized algorithms
Interactive proofs and the hardness of approximating cliques
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Derandomizing polynomial identity tests means proving circuit lower bounds
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Primality and identity testing via Chinese remaindering
Journal of the ACM (JACM)
Splitters and near-optimal derandomization
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Deterministic polynomial identity testing in non-commutative models
Computational Complexity
Improved algorithms for path, matching, and packing problems
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Faster Algebraic Algorithms for Path and Packing Problems
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Finding paths of length k in O∗(2k) time
Information Processing Letters
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part I
The complexity of testing monomials in multivariate polynomials
COCOA'11 Proceedings of the 5th international conference on Combinatorial optimization and applications
The complexity of testing monomials in multivariate polynomials
COCOA'11 Proceedings of the 5th international conference on Combinatorial optimization and applications
FAW-AAIM'12 Proceedings of the 6th international Frontiers in Algorithmics, and Proceedings of the 8th international conference on Algorithmic Aspects in Information and Management
Journal of Combinatorial Optimization
On testing monomials in multivariate polynomials
Theoretical Computer Science
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This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its sum-product expansion. The complexity aspects of this problem and its variants have been investigated in our first paper by Chen and Fu (2010), laying a foundation for further study. In this paper, we present two pairs of algorithms. First, we prove that there is a randomized O*(pk) time algorithm for testing p-monomials in an n-variate polynomial of degree k represented by an arithmetic circuit, while a deterministic O*((6.4p)k) time algorithm is devised when the circuit is a formula, here p is a given prime number. Second, we present a deterministic O*(2k) time algorithm for testing multilinear monomials in ΠmΣ2Πt×ΠkΣ3 polynomials, while a randomized O*(1.5k) algorithm is given for these polynomials. Finally, we prove that testing some special types of multilinear monomial is W[1]-hard, giving evidence that testing for specific monomials is not fixed-parameter tractable.