Hybrid Rounding Techniques for Knapsack Problems

TL;DR

This paper introduces hybrid rounding methods for knapsack problems, leading to new approximation schemes that significantly improve computational efficiency while maintaining solution accuracy.

Contribution

It presents novel hybrid rounding techniques and develops linear-time PTAS and FPTAS for knapsack problems, improving upon previous polynomial bounds.

Findings

Linear-time PTAS and FPTAS for knapsack problems

Hybrid rounding techniques enhance approximation quality

Significant reduction in computational complexity

Abstract

We address the classical knapsack problem and a variant in which an upper bound is imposed on the number of items that can be selected. We show that appropriate combinations of rounding techniques yield novel and powerful ways of rounding. As an application of these techniques, we present a linear-storage Polynomial Time Approximation Scheme (PTAS) and a Fully Polynomial Time Approximation Scheme (FPTAS) that compute an approximate solution, of any fixed accuracy, in linear time. This linear complexity bound gives a substantial improvement of the best previously known polynomial bounds.