LIPIcs.ISAAC.2019.38.pdf
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On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials
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30th International Symposium on Algorithms and Computation (ISAAC 2019)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Symposium on Algorithms and Computation (ISAAC) - License:
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- Publication Date: 2019-11-28
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V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 38:1-38:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.38
https://doi.org/10.4230/LIPIcs.ISAAC.2019.38
Abstract
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^*(binom{n}{downarrow k/2})-size ABP construction for noncommutative permanent polynomial of k x n symbolic matrix. We obtain this via an explicit ABP construction of size O^*(binom{n}{downarrow 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 x n symbolic matrix.
- In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is #W[1]-hard.
Subject Classification
ACM Subject Classification
- Theory of computation → Algebraic complexity theory
- Theory of computation
Keywords
- Determinant
- Permanent
- Parameterized Complexity
- Branching Programs
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References
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