David Hartman and Milan Hladík. Complexity of computing interval matrix powers for special classes of matrices. Appl. Math., 65(5):645–663, 2020.
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Computing powers of interval matrices is a computationally hard problem. Indeed, it is NP-hard even when the exponent is 3 and the matrices only have interval components in one row and one column. Motivated by this result, we consider special types of interval matrices where the interval components occupy specific positions. We show that computing the third power of matrices with only one column occupied by interval components can be solved in cubic time; so the asymptotic time complexity is the same as for the real case (considering the textbook matrix product method). We further show that for a fixed exponent k and for each interval matrix (of an arbitrary size) whose kth power has components that can be expressed as polynomials in a fixed number of interval variables, the computation of the kth power is polynomial up to a given accuracy. Polynomiality is shown by using the Tarski method of quantifier elimination. This result is used to show the polynomiality of computing the cube of interval band matrices, among others. Additionally, we study parametric matrices and prove NP-hardness already for their squares. We also describe one specific class of interval parametric matrices that can be squared by a polynomial algorithm.
@article{HarHla2020a,
author = "David Hartman and Milan Hlad\'{\i}k",
title = "Complexity of computing interval matrix powers for special classes of matrices",
journal = "Appl. Math.",
fjournal = "Applications of Mathematics",
volume = "65",
number = "5",
pages = "645-663",
year = "2020",
doi = "10.21136/AM.2020.0379-19",
issn = "1572-9109",
url = "https://doi.org/10.21136/AM.2020.0379-19",
bib2html_dl_html = "https://link.springer.com/article/10.21136/AM.2020.0379-19",
bib2html_dl_pdf = "https://rdcu.be/b8vtG",
abstract = "Computing powers of interval matrices is a computationally hard problem. Indeed, it is NP-hard even when the exponent is 3 and the matrices only have interval components in one row and one column. Motivated by this result, we consider special types of interval matrices where the interval components occupy specific positions. We show that computing the third power of matrices with only one column occupied by interval components can be solved in cubic time; so the asymptotic time complexity is the same as for the real case (considering the textbook matrix product method). We further show that for a fixed exponent k and for each interval matrix (of an arbitrary size) whose kth power has components that can be expressed as polynomials in a fixed number of interval variables, the computation of the kth power is polynomial up to a given accuracy. Polynomiality is shown by using the Tarski method of quantifier elimination. This result is used to show the polynomiality of computing the cube of interval band matrices, among others. Additionally, we study parametric matrices and prove NP-hardness already for their squares. We also describe one specific class of interval parametric matrices that can be squared by a polynomial algorithm.",
keywords = "Matrix power; Interval matrix; Interval computations; NP-hardness",
}
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