  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
$\Omega(2^{n/2}/\sqrt n)=\Omega(2^{0.5n}/\sqrt n)$ and
$\Omega((1/\epsilon)^{1/3.17})\approx\Omega((1/\epsilon)^{0.315})$(for
approximation ratio $1-\epsilon$) 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
$\Omega(2^{(2-\epsilon)n/3}/\sqrt{n})\approx \Omega(2^{0.66n}/\sqrt{n})$ and
$\Omega((1/\epsilon)^{1/2.38})\approx\Omega((1/\epsilon)^{0.420})$, and
proposed as an open question what is the best achievable lower bounds for
knapsack under adaptive BT models.
