Improved Exponential Time Lower Bound of Knapsack Problem under BT model

TL;DR

This paper improves the exponential time lower bounds for the Knapsack problem under the adaptive BT model, narrowing the gap between known bounds and raising open questions about optimal limits.

Contribution

It refines existing lower bounds for the Knapsack problem in the adaptive BT model and highlights open problems for future research.

Findings

Lower bounds improved to approximately 2^{0.66n}/√n

Approximate lower bounds for ratio 1-ε improved to about (1/ε)^{0.420}

Open question posed on the optimal bounds achievable

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 $\Omega(2^{n/2}/\sqrt n)=\Omega(2^{0.5n}/\sqrt n)$ 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.