Year: 2006
Type: preprint
Abstract: M.Alekhnovich et al. recently have proposed a model of algorithms, called BT model, which covers Greedy, Backtrack and Simple Dynamic Programming methods and can be further divided into fixed, adaptive and fully adaptive three kinds, and have proved exponential time lower bounds of exact and approximation algorithms under adaptive BT model for Knapsack problem which are $Ω(2^{n/2}/\sqrt n)=Ω(2^{0.5n}/\sqrt n)$ and $Ω((1/ε)^{1/3.17})\approxΩ((1/ε)^{0.315})$(for approximation ratio $1-ε$) respectively (M. Alekhovich, A. Borodin, J. Buresh-Oppenheim, R. Impagliazzo, A. Magen, and T. Pitassi, Toward a Model for Backtracking and Dynamic Programming, \emph{Proceedings of Twentieth Annual IEEE Conference on Computational Complexity}, pp308-322, 2005). In this note, we slightly improved their lower bounds to $Ω(2^{(2-ε)n/3}/\sqrt{n})\approx Ω(2^{0.66n}/\sqrt{n})$ and $Ω((1/ε)^{1/2.38})\approxΩ((1/ε)^{0.420})$, and proposed as an open question what is the best achievable lower bounds for knapsack under adaptive BT models.
Source: arXiv (Cornell University)
Authors: Li, Xin, Liu, Tian, Peng, Han, Sun, Hongtao, Zhu, Jiaqi
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2907929595