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Zhixiang Chen - One of the best experts on this subject based on the ideXlab platform.

  • on testing Monomials in multivariate polynomials
    Theoretical Computer Science, 2013
    Co-Authors: Zhixiang Chen, Yang Liu, Robert T Schweller
    Abstract:

    This paper presents a summary of our initial work on developing a theory of testing Monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its sum-product expansion. The complexity aspects of this problem and its variants are investigated with two objectives. One is to understand how this problem relates to critical problems in complexity, and if so to what extent. The other is to exploit possibilities of applying algebraic properties of polynomials to the study of those problems. A series of results about @[email protected]@P and @[email protected] polynomials is obtained in this paper, laying a basis for further study along this line. Several randomized and deterministic algorithms are devised for testing multilinear Monomials or p-Monomials in certain respective types of polynomials, where p is prime.

  • approximating multilinear monomial coefficients and maximum multilinear Monomials in multivariate polynomials
    Journal of Combinatorial Optimization, 2013
    Co-Authors: Zhixiang Chen
    Abstract:

    This paper is our third step towards developing a theory of testing Monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear Monomials; and (2) how to find a maximum multilinear monomial when the input is a ?Σ? polynomial. We first prove that the first problem is #P-hard and then devise a O ?(3 n s(n)) upper bound for this problem for any polynomial represented by an arithmetic circuit of size s(n). Later, this upper bound is improved to O ?(2 n ) for ?Σ? polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for ?Σ polynomials. On the negative side, we prove that, even for ?Σ? polynomials with terms of degree ?2, the first problem cannot be approximated at all for any approximation factor ?1, nor "weakly approximated" in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time ?-approximation algorithm for ?Σ? polynomials with terms of degrees no more a constant ??2. On the inapproximability side, we give a n (1??)/2 lower bound, for any ?>0, on the approximation factor for ?Σ? polynomials. When terms in these polynomials are constrained to degrees ?2, we prove a 1.0476 lower bound, assuming P?NP; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.

  • approximating multilinear monomial coefficients and maximum multilinear Monomials in multivariate polynomials
    Conference on Combinatorial Optimization and Applications, 2010
    Co-Authors: Zhixiang Chen
    Abstract:

    This paper is our third step towards developing a theory of testing Monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear Monomials; and (2) how to find a maximum multilinear monomial when the input is a ΠΣΠ polynomial. We first prove that the first problem is #P-hard and then devise a O*(3n s(n)) upper bound for this problem for any polynomial represented by an arithmetic circuit of size s(n). Later, this upper bound is improved to O*(2n) for ΠΣΠ polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for ΠΣ polynomials. On the negative side, we prove that, even for ΠΣΠ polynomials with terms of degree ≤ 2, the first problem cannot be approximated at all for any approximation factor ≥ 1, nor "weakly approximated" in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time λ-approximation algorithm for ΠΣΠ polynomials with terms of degrees no more a constant λ ≥ 2. On the inapproximability side, we give a n(1-e)/2 lower bound, for any e > 0, on the approximation factor for ΠΣΠ polynomials. When the degrees of the terms in these polynomials are constrained as le; 2, we prove a 1.0476 lower bound, assuming P ≠ NP; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.

  • algorithms for testing Monomials in multivariate polynomials
    arXiv: Computational Complexity, 2010
    Co-Authors: Zhixiang Chen, Yang Liu, Robert T Schweller
    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 $O^*(6.4^k + p^k)$ time algorithm is devised when the circuit is a formula, here $p$ is a given prime number. Second, we present a deterministic $O^*(2^k)$ time algorithm for testing multilinear Monomials in $\Pi_m\Sigma_2\Pi_t\times \Pi_k\Pi_3$ polynomials, while a randomized $O^*(1.5^k)$ algorithm is given for these polynomials. The first algorithm extends the recent work by Koutis (2008) and Williams (2009) on testing multilinear Monomials. Group algebra is exploited in the algorithm designs, in corporation with the randomized polynomial identity testing over a finite field by Agrawal and Biswas (2003), the deterministic noncommunicative polynomial identity testing by Raz and Shpilka (2005) and the perfect hashing functions by Chen {\em at el.} (2007). 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.

  • approximating multilinear monomial coefficients and maximum multilinear Monomials in multivariate polynomials
    arXiv: Computational Complexity, 2010
    Co-Authors: Zhixiang Chen
    Abstract:

    This paper is our third step towards developing a theory of testing Monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear Monomials; and (2) how to find a maximum multilinear monomial when the input is a $\Pi\Sigma\Pi$ polynomial. We first prove that the first problem is \#P-hard and then devise a $O^*(3^ns(n))$ upper bound for this problem for any polynomial represented by an arithmetic circuit of size $s(n)$. Later, this upper bound is improved to $O^*(2^n)$ for $\Pi\Sigma\Pi$ polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for $\Pi\Sigma$ polynomials. On the negative side, we prove that, even for $\Pi\Sigma\Pi$ polynomials with terms of degree $\le 2$, the first problem cannot be approximated at all for any approximation factor $\ge 1$, nor {\em "weakly approximated"} in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time $\lambda$-approximation algorithm for $\Pi\Sigma\Pi$ polynomials with terms of degrees no more a constant $\lambda \ge 2$. On the inapproximability side, we give a $n^{(1-\epsilon)/2}$ lower bound, for any $\epsilon >0,$ on the approximation factor for $\Pi\Sigma\Pi$ polynomials. When terms in these polynomials are constrained to degrees $\le 2$, we prove a $1.0476$ lower bound, assuming $P\not=NP$; and a higher $1.0604$ lower bound, assuming the Unique Games Conjecture.

Stefan Mengel - One of the best experts on this subject based on the ideXlab platform.

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Prasad Raghavendra - One of the best experts on this subject based on the ideXlab platform.

  • agnostic learning of Monomials by halfspaces is hard
    SIAM Journal on Computing, 2012
    Co-Authors: Vitaly Feldman, Venkatesan Guruswami, Prasad Raghavendra
    Abstract:

    We prove the following strong hardness result for learning: Given a distribution of labeled examples from the hypercube such that there exists a monomial consistent with $(1-\epsilon)$ of the examples it is NP-hard to find a halfspace that is correct on $(1/2+\epsilon)$ of the examples for arbitrary constants $\epsilon>0$. In learning theory terms, weak agnostic learning of Monomials is hard, even if one is allowed to output a hypothesis from the much bigger concept class of halfspaces. This hardness result subsumes a long line of previous results, including two recent hardness results for the proper learning of Monomials and halfspaces. As an immediate corollary of our result we show that weak agnostic learning of decision lists is NP-hard. Our techniques are quite different from previous hardness proofs for learning. We define distributions on positive and negative examples for Monomials whose first few moments match. We use the invariance principle to argue that regular halfspaces (all of whose coeffic...

  • agnostic learning of Monomials by halfspaces is hard
    arXiv: Computational Complexity, 2010
    Co-Authors: Vitaly Feldman, Venkatesan Guruswami, Prasad Raghavendra
    Abstract:

    We prove the following strong hardness result for learning: Given a distribution of labeled examples from the hypercube such that there exists a monomial consistent with $(1-\eps)$ of the examples, it is NP-hard to find a halfspace that is correct on $(1/2+\eps)$ of the examples, for arbitrary constants $\eps > 0$. In learning theory terms, weak agnostic learning of Monomials is hard, even if one is allowed to output a hypothesis from the much bigger concept class of halfspaces. This hardness result subsumes a long line of previous results, including two recent hardness results for the proper learning of Monomials and halfspaces. As an immediate corollary of our result we show that weak agnostic learning of decision lists is NP-hard. Our techniques are quite different from previous hardness proofs for learning. We define distributions on positive and negative examples for Monomials whose first few moments match. We use the invariance principle to argue that regular halfspaces (all of whose coefficients have small absolute value relative to the total $\ell_2$ norm) cannot distinguish between distributions whose first few moments match. For highly non-regular subspaces, we use a structural lemma from recent work on fooling halfspaces to argue that they are ``junta-like'' and one can zero out all but the top few coefficients without affecting the performance of the halfspace. The top few coefficients form the natural list decoding of a halfspace in the context of dictatorship tests/Label Cover reductions. We note that unlike previous invariance principle based proofs which are only known to give Unique-Games hardness, we are able to reduce from a version of Label Cover problem that is known to be NP-hard. This has inspired follow-up work on bypassing the Unique Games conjecture in some optimal geometric inapproximability results.

  • agnostic learning of Monomials by halfspaces is hard
    Foundations of Computer Science, 2009
    Co-Authors: Vitaly Feldman, Venkatesan Guruswami, Prasad Raghavendra
    Abstract:

    We prove the following strong hardness result for learning: Given a distribution on labeled examples from the hypercube such that there exists a monomial (or conjunction) consistent with (1-e)-fraction of the examples, it is NP-hard to find a halfspace that is correct on ( 1/2 +e)-fraction of the examples, for arbitrary constant e ≫ 0. In learning theory terms, weak agnostic learning of Monomials by halfspaces is NP-hard. This hardness result bridges between and subsumes two previous results which showed similar hardness results for the proper learning of Monomials and halfspaces. As immediate corollaries of our result, we give the first optimal hardness results for weak agnostic learning of decision lists and majorities. Our techniques are quite different from previous hardness proofs for learning. We use an invariance principle and sparse approximation of halfspaces from recent work on fooling halfspaces to give a new natural list decoding of a halfspace in the context of dictatorship tests/label cover reductions. In addition, unlike previous invariance principle based proofs which are only known to give Unique Games hardness, we give a reduction from a smooth version of Label Cover that is known to be NP-hard.

Guillaume Malod - One of the best experts on this subject based on the ideXlab platform.