On Exponential Time Lower Bound of Knapsack under Backtracking

TL;DR

This paper improves the exponential time lower bound for solving the Knapsack problem under the adaptive BT model, demonstrating that even advanced algorithms require exponential time.

Contribution

It provides a tighter exponential lower bound for the Knapsack problem within the adaptive BT model, enhancing previous bounds with optimized parameters.

Findings

Lower bound improved to approximately Ω(2^{0.69n}/√n)

Demonstrates inherent exponential complexity of Knapsack under the BT model

Uses refined techniques to optimize parameters for the lower bound

Abstract

M.Aleknovich et al. have recently proposed a model of algorithms, called BT model, which generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model by Woeginger. BT model can be further divided into three kinds of fixed, adaptive and fully adaptive ones. They have proved exponential time lower bounds of exact and approximation algorithms under adaptive BT model for Knapsack problem. Their exact lower bound is $\Omega(2^{0.5n}/\sqrt{n})$, in this paper, we slightly improve the exact lower bound to about $\Omega(2^{0.69n}/\sqrt{n})$, by the same technique, with related parameters optimized.