Fast Exact Algorithms Using Hadamard Product of Polynomials

Arvind, V.; Chatterjee, Abhranil; Datta, Rajit; Mukhopadhyay, Partha

doi:10.48550/arxiv.1807.04496

Cited by 2 publications

(11 citation statements)

References 18 publications

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“…Brand et al have given the first FPT algorithm for multilinear monomial detection in the case of general circuit with run time randomized O * (4.32 k ) [8]. Recently, this problem has also been studied using the Hadamard product [3] of the given polynomial with the elementary symmetric polynomial (and differently using apolar bilinear forms [23]). Our proof of Theorem 1.5 shows that checking membership of f in the ideal x e 1 1 , .…”

Section: Our Resultsmentioning

confidence: 99%

“…The proof easily follows from our recent work [3]. We include a self-contained proof in the appendix (Section B).…”

Section: Lemma 42 Given a Circuit C Of Size S Computing A Polynomial ...mentioning

confidence: 94%

“…The proof uses the Hadamard product of polynomials and a connection to noncommutative computation. This builds on our recent work [3]. We include Section B in the appendix to provide the background.…”

Section: Proof Of Theorem 15mentioning

confidence: 99%

“…Then we show that we can compute Hadamard product of any two polynomials in time roughly linear in the product of the size of the circuits when one of the polynomials is given by a diagonal circuit as input. Finally the last part of the proof is a randomized construction of a homogeneous degree k diagonal circuit of top fain-in roughly O * (4.08 k ) that computes a polynomial weakly equivalent 3 to the polynomial g with constant probability.…”

Section: Proof Of Theorem 15mentioning

confidence: 99%

“…) k ) uniformly at 3 Two polynomials f and g are said to be weakly equivalent if they share the same set of monomials. random.…”

Section: Lemma 42 Given a Circuit C Of Size S Computing A Polynomial ...mentioning

confidence: 99%

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Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators

Arvind

Chatterjee

Datta

et al. 2018

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Let F[X] be the polynomial ring over the variables X = {x 1 , x 2 , . . . , x n }. An ideal I = p 1 (x 1 ), . . . , p n (x n ) generated by univariate polynomials {p i (x i )} n i=1 is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results.• Let f (X) ∈ F[ℓ 1 , . . . , ℓ r ] be a (low rank) polynomial given by an arithmetic circuit where ℓ i : 1 ≤ i ≤ r are linear forms, and I = p 1 (x 1 ), . . . , p n (x n ) be a univariate ideal. Given α ∈ F n , the (unique) remainder f (X) (mod I) can be evaluated at α in deterministic time d O(r) • poly(n), where d = max{deg(f ), deg(p 1 ) . . . , deg(p n )}. This yields an n O(r) algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields an n O(r) algorithm for evaluating the permanent of a n × n matrix of rank r, over any field F. Over Q, an algorithm of similar run time for low rank permanent is due to Barvinok [6] via a different technique.• Let f (X) ∈ F[X] be given by an arithmetic circuit of degree k (k treated as fixed parameter) and I = p 1 (x 1 ), . . . , p n (x n ) . We show that in the special case when I = x e1 1 , . . . , x en n , we obtain a randomized O * (4.08 k ) algorithm that uses poly(n, k) space.• Given f (X) ∈ F[X] by an arithmetic circuit and I = p 1 (x 1 ), . . . , p k (x k ) , membership testing is W[1]-hard, parameterized by k. The problem is MINI[1]-hard in the special case when I = x e1 1 , . . . , x e k k .

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Section: Our Resultsmentioning

confidence: 99%

“…The proof easily follows from our recent work [3]. We include a self-contained proof in the appendix (Section B).…”

Section: Lemma 42 Given a Circuit C Of Size S Computing A Polynomial ...mentioning

confidence: 94%

Section: Proof Of Theorem 15mentioning

confidence: 99%

Section: Proof Of Theorem 15mentioning

confidence: 99%

“…) k ) uniformly at 3 Two polynomials f and g are said to be weakly equivalent if they share the same set of monomials. random.…”

Section: Lemma 42 Given a Circuit C Of Size S Computing A Polynomial ...mentioning

confidence: 99%

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Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators

Arvind

Chatterjee

Datta

et al. 2018

Preprint

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On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials

Arvind

Chatterjee

Datta

et al. 2019

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We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the rectangular symbolic matrix in both commutative and noncommutative settings. The main results are:We show an explicit O * ( n ↓k/2 )-size ABP construction for noncommutative permanent polynomial of k×n symbolic matrix. We obtain this via an explicit ABP construction of size O * ( n ↓k/2 ) for S * n,k , noncommutative symmetrized version of the elementary symmetric polynomial S n,k . We obtain an explicit O * (2 k )-size ABP construction for the commutative rectangular determinant polynomial of the k × n symbolic matrix. In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is #W[1]-hard.

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Fast Exact Algorithms Using Hadamard Product of Polynomials

Abstract: In this paper we develop an efficient procedure for computing a (scaled) Hadamard product for commutative polynomials. This serves as a tool to obtain faster algorithms for several problems. Our main algorithmic results include the following:

Cited by 2 publications

References 18 publications

Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators

Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators

On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials

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