Abstract
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 *(p k) 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 Π m Σ2Π t ×Π k Σ3 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.
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Chen, Z., Fu, B., Liu, Y., Schweller, R. (2011). Algorithms for Testing Monomials in Multivariate Polynomials. In: Wang, W., Zhu, X., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2011. Lecture Notes in Computer Science, vol 6831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22616-8_2
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DOI: https://doi.org/10.1007/978-3-642-22616-8_2
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