Arithmetic Circuits and the Hadamard Product of Polynomials

Arvind, Vikraman; Joglekar, Pushkar S.; Srinivasan, Srikanth

doi:10.4230/lipics.fsttcs.2009.2304

Cited by 2 publications

(8 citation statements)

References 10 publications

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“…Broadly, it works as follows: we transform polynomials f and g to suitable noncommutative polynomials. We compute their (noncommutative) Hadamard product efficiently [3,5], and we finally recover the scaled commutative Hadamard product f…”

Section: Definitionmentioning

confidence: 99%

“…Notice that, by the proof of Lemma 16, it is easy to see that C i ( a) can be computed deterministically in time 2 k •poly(n, s) time and poly(n, k) space. 3 To improve the run time from O * ((2e) k ) to O * (4.32 k ), we can use the idea of Hüffner et al [12] 4 . The key idea is that, using more than k colors we would reduce the number of colorings and hence the number of ΠΣ circuits, but it would increase the formal degree of each P i .…”

Section: Fast Exact Algorithms Using Hadamard Product Of Polynomialsmentioning

confidence: 99%

“…The Hadamard product (also known as Schur product) generally refers to Hadamard product of matrices and is used in matrix analysis. We consider the Hadamard product of polynomials (e.g., see [3]). Given polynomials f, g ∈ F[X], their Hadamard product is defined as f…”

Section: Introductionmentioning

confidence: 99%

“…The Hadamard product is a useful tool in noncommutative computation [3,5]. A contribution of the present paper is to develop an efficient way to implement Hadamard product in the commutative setting which is useful for designing FPT and exact algorithms.…”

Section: Introductionmentioning

confidence: 99%

“…We apply the Hadamard product of polynomials in the setting of commutative computation. This is achieved by combining earlier ideas [3,5] with a symmetrization trick shown in Section 2. We then use it to design efficient algorithms for (k,n)-MLC, k-MMD and related problems.…”

Section: Introductionmentioning

confidence: 99%

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Fast Exact Algorithms Using Hadamard Product of Polynomials

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Let C be an arithmetic circuit of poly(n) size given as input that computes a polynomial f ∈ F[X], where X = {x1, x2, . . . , xn} 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 [13,22,14].(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 • 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 [14, ICALP'09]. As corollaries, we show O * ( n ↓k/2

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Section: Definitionmentioning

confidence: 99%

Section: Fast Exact Algorithms Using Hadamard Product Of Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fast Exact Algorithms Using Hadamard Product of Polynomials

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On the hardness of the noncommutative determinant

Arvind

Srinivasan

2010

Proceedings of the Forty-Second ACM Symposium on Theory of Computing

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In this paper we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below:• We show that if the noncommutative determinant polynomial has small noncommutative arithmetic circuits then so does the noncommutative permanent. Consequently, the commutative permanent polynomial has small commutative arithmetic circuits.• For any field F we show that computing the n × n permanent over F is polynomial-time reducible to computing the 2n × 2n (noncommutative) determinant whose entries are O(n 2 ) × O(n 2 ) matrices over the field F.• We also derive as a consequence that computing the n × n permanent over nonnegative rationals is polynomial-time reducible to computing the noncommutative determinant over Clifford algebras of n O(1) dimension.Our techniques are elementary and use primarily the notion of the Hadamard Product of noncommutative polynomials.

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Arithmetic Circuits and the Hadamard Product of Polynomials

Cited by 2 publications

References 10 publications

Fast Exact Algorithms Using Hadamard Product of Polynomials

Fast Exact Algorithms Using Hadamard Product of Polynomials

On the hardness of the noncommutative determinant

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