License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2019.9
URN: urn:nbn:de:0030-drops-115711
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11571/
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Arvind, V. ; Chatterjee, Abhranil ; Datta, Rajit ; Mukhopadhyay, Partha

Fast Exact Algorithms Using Hadamard Product of Polynomials

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Abstract

Let C be an arithmetic circuit of poly(n) size given as input that computes a polynomial f in F[X], where X={x_1,x_2,...,x_n} and F is any field where the field arithmetic can be performed efficiently. We obtain new algorithms for the following two problems first studied by Koutis and Williams [Ioannis Koutis, 2008; Ryan Williams, 2009; Ioannis Koutis and Ryan Williams, 2016].
- (k,n)-MLC: Compute the sum of the coefficients of all degree-k multilinear monomials in the polynomial f.
- k-MMD: Test if there is a nonzero degree-k multilinear monomial in the polynomial f.
Our algorithms are based on the fact that the Hadamard product f o S_{n,k}, is the degree-k multilinear part of f, where S_{n,k} is the k^{th} elementary symmetric polynomial.
- For (k,n)-MLC problem, we give a deterministic algorithm of run time O^*(n^(k/2+c log k)) (where c is a constant), answering an open question of Koutis and Williams [Ioannis Koutis and Ryan Williams, 2016]. As corollaries, we show O^*(binom{n}{downarrow k/2})-time exact counting algorithms for several combinatorial problems: k-Tree, t-Dominating Set, m-Dimensional k-Matching.
- For k-MMD problem, we give a randomized algorithm of run time 4.32^k * poly(n,k). Our algorithm uses only poly(n,k) space. This matches the run time of a recent algorithm [Cornelius Brand et al., 2018] for k-MMD which requires exponential (in k) space.
Other results include fast deterministic algorithms for (k,n)-MLC and k-MMD problems for depth three circuits.

BibTeX - Entry

@InProceedings{arvind_et_al:LIPIcs:2019:11571,
  author =	{V. Arvind and Abhranil Chatterjee and Rajit Datta and Partha Mukhopadhyay},
  title =	{{Fast Exact Algorithms Using Hadamard Product of Polynomials}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{9:1--9:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Arkadev Chattopadhyay and Paul Gastin},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2019/11571},
  URN =		{urn:nbn:de:0030-drops-115711},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.9},
  annote =	{Keywords: Hadamard Product, Multilinear Monomial Detection and Counting, Rectangular Permanent, Symmetric Polynomial}
}

Keywords: Hadamard Product, Multilinear Monomial Detection and Counting, Rectangular Permanent, Symmetric Polynomial
Collection: 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)
Issue Date: 2019
Date of publication: 04.12.2019

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