Complexity of Computing Interval Matrix Powers for Special Classes of Matrices

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F20%3A00532104" target="_blank" >RIV/67985807:_____/20:00532104 - isvavai.cz</a>
Alternative codes found
RIV/00216208:11320/20:10419270
Result on the web
<a href="http://hdl.handle.net/11104/0310705" target="_blank" >http://hdl.handle.net/11104/0310705</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.21136/AM.2020.0379-19" target="_blank" >10.21136/AM.2020.0379-19</a>

Result language
angličtina
Original language name
Complexity of Computing Interval Matrix Powers for Special Classes of Matrices
Original language description
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 $k$th power has components that can be expressed as polynomials in a fixed number of interval variables, the computation of the $k$th 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.
Czech name
—
Czech description
—

Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA18-04735S" target="_blank" >GA18-04735S: Novel approaches for relaxation and approximation techniques in deterministic global optimization</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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