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International Computing and Combinatorics Conference

COCOON 2019: Computing and Combinatorics pp 178–192Cite as

On Proving Parameterized Size Lower Bounds for Multilinear Algebraic Models

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11653))

Abstract

We consider the problem of obtaining parameterized lower bounds for the size of arithmetic circuits computing polynomials with the degree of the polynomial as the parameter. In particular, we consider the following special classes of multilinear algebraic branching programs: (1) Read Once Oblivious Algebraic Branching Programs (ROABPs); (2) Strict interval branching programs; and (3) Sum of read once formulas with restricted ordering.

We obtain parameterized lower bounds (i.e., \(n^{\varOmega (t(k))}\) lower bound for some function t of k) on the size of the above models computing a multilinear polynomial that can be computed by a depth four circuit of size \(g(k) n^{O(1)}\) for some computable function g.

Our proof is an adaptation of the existing techniques to the parameterized setting. The main challenge we address is the construction of hard parameterized polynomials. In fact, we show that there are polynomials computed by depth four circuits of small size (in the parameterized sense), but have high rank of the partial derivative matrix.

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Authors and Affiliations

  1. Indian Institute of Technology Madras, Chennai, 600036, India

    Purnata Ghosal & B. V. Raghavendra Rao

Authors
  1. Purnata Ghosal
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  2. B. V. Raghavendra Rao
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Correspondence to Purnata Ghosal .

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Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. Xidian University, Xi'an, China

    Zhenhua Duan

  3. Xidian University, Xi'an, China

    Cong Tian

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Ghosal, P., Raghavendra Rao, B.V. (2019). On Proving Parameterized Size Lower Bounds for Multilinear Algebraic Models. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_15

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  • DOI: https://doi.org/10.1007/978-3-030-26176-4_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26175-7

  • Online ISBN: 978-3-030-26176-4

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