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Fast Exact Algorithms Using Hadamard Product of Polynomials

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Abstract

Let C be an arithmetic circuit of size s, given as input that computes a polynomial \(f\in {\mathbb {F}}[x_1,x_2,\ldots ,x_n]\), where \({\mathbb {F}}\) is a finite field or the field of rationals. Using the Hadamard product of polynomials, we obtain new algorithms for the following two problems first studied by Koutis and Williams (Faster algebraic algorithms for path and packing problems, 2008, https://doi.org/10.1007/978-3-540-70575-8_47; ACM Trans Algorithms 12(3):31:1–31:18, 2016, https://doi.org/10.1145/2885499; Inf Process Lett 109(6):315–318, 2009, https://doi.org/10.1016/j.ipl.2008.11.004):

  • \({{{(\textit{k,n}){-}\mathrm{M{L}\normalsize {C}}}}}\): is the problem of computing the sum of the coefficients of all degree-k multilinear monomials in the polynomial f. We obtain a deterministic algorithm of running time \({n\atopwithdelims (){\downarrow k/2}}\cdot n^{O(\log k)}\cdot s^{O(1)}\). This improvement over the \(O(n^k)\) time brute-force search algorithm answers positively a question of Koutis and Williams (2016). As applications, we give exact counting algorithms, faster than brute-force search, for counting the number of copies of a tree of size k in a graph, and also the problem of exact counting of m-dimensional k-matchings.

  • \({{{\textit{k}{-}\mathrm{M{M}\normalsize {D}}}}}\): is the problem of checking if there is a degree-k multilinear monomial in the polynomial f with non-zero coefficient. We obtain a randomized algorithm of running time \(O(4.32^k\cdot n^{O(1)})\). Additionally, our algorithm is polynomial space bounded.

Other results include fast deterministic algorithms for \({{{(\textit{k,n}){-}\mathrm{M{L}\normalsize {C}}}}}\) and \({{{\textit{k}{-}\mathrm{M{M}\normalsize {D}}}}}\) problems for depth three circuits.

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Notes

  1. In this formulation, of course, the problem is a generalization of counting the degree-k multilinear monomials.

  2. The \(O^*\) notation suppresses factors polynomial in the input size.

  3. Throughout, we use \(y_i\) to denote noncommuting variables associated with the \(x_i\).

  4. This terminology is in keeping with a seminal paper’s in the field [1] which introduced color coding. However, it should be clear that this notion of coloring has nothing to do with graph colorings.

  5. Since the syntactic degree of the circuit is not bounded here, and if we have to account for the bit level complexity (over \({\mathbb {Z}}\)) of the scalars generated in the intermediate stage we may get field elements whose bit level complexity is exponential in the input size. So, a standard technique is to take a random prime of polynomial bit-size and evaluate the circuit modulo that prime.

  6. By 1.3k and 0.3k, we mean the integers \(\lceil 1.3k\rceil \) and \(\lceil 0.3k\rceil \), respectively.

  7. A polynomial g is positively weakly equivalent to f if it has the same set of nonzero monomials, with any positive coefficients.

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Acknowledgements

We are grateful to the referees for their comments and suggestions that have helped us improve the presentation. We thank anonymous reviewers of FSTTCS 2019 for their comments on an earlier version of this paper.

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Authors and Affiliations

  1. Institute of Mathematical Sciences (HBNI), Chennai, India

    V. Arvind & Abhranil Chatterjee

  2. Chennai Mathematical Institute, Chennai, India

    Rajit Datta & Partha Mukhopadhyay

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An earlier version was presented at the FSTTCS 2019 conference [3] (arxiv version [2])

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Arvind, V., Chatterjee, A., Datta, R. et al. Fast Exact Algorithms Using Hadamard Product of Polynomials. Algorithmica 84, 436–463 (2022). https://doi.org/10.1007/s00453-021-00900-0

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