### Abstract

Tractability of multivariate problems studies their complexity with respect to the number of variables, d, and the accuracy of the solution ε. Different types of tractability have been used, such as polynomial tractability and weak tractability and others. These tractability types, however, do not express the complexity with respect to the number of bits of accuracy. We introduce two new tractability types, polylog tractability and ln ^{κ}-weak tractability. A problem is polylog tractable iff its complexity is polynomial in d and in lnε-^{1}, while a problem is ln^{κ}-weakly tractable iff its complexity is not exponential in d and ln^{κ}ε-^{1}, for some κâ‰ ¥ 1. We study general multivariate problems and multivariate tensor product problems. We provide necessary and sufficient conditions for the respective tractability types. Moreover, we show that a multivariate tensor product problem cannot be polylog tractable and cannot be ln^{1}-weakly tractable (i.e., with κ=1) unless it is trivial.

Original language | English (US) |
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Pages (from-to) | 604-619 |

Number of pages | 16 |

Journal | Journal of Complexity |

Volume | 30 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2014 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- Mathematics(all)
- Control and Optimization
- Applied Mathematics

### Keywords

- Complexity
- Multivariate problem
- Tractability

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## Cite this

*Journal of Complexity*,

*30*(5), 604-619. https://doi.org/10.1016/j.jco.2014.03.001