An O(nlogn) approximate knapsack algorithm

TL;DR

This paper introduces a modified dynamic programming algorithm for large 0/1 knapsack problems that achieves near-linear time complexity, high accuracy, and fast processing speeds on standard hardware.

Contribution

It presents a novel O(nlogn) approximate algorithm for 0/1 knapsack problems with predictable error bounds and exceptional computational efficiency.

Findings

Achieves average maximum fractional error of 1e-4 for k=1,000

Achieves average maximum fractional error of 1e-7 for k=100,000

Processes 1,000 problems in 1 millisecond and 1 million problems in 2 seconds

Abstract

A modified dynamic programming algorithm rapidly and accurately solves large 0/1 knapsack problems. It has computational O(nlogn), space O(nlogn) and predictable maximum error. Experimentally it's accuracy increases faster than linearly with the solution size k. Problems with k=1e3 are solved with an average maximum fractional error of 1e-4 and problems with k=1e5 with an average maximum fractional error of 1e-7. The algorithm runs in constant time for all problems with a given n. On a common desktop computer the algorithm processes n=1e3 problems in 1e-3 seconds and n=1e6 problems in 2 seconds.