An Improved Unbounded-DP Algorithm for the Unbounded Knapsack Problem with Bounded Coefficients

• Published in: Mathematics (Basel) Vol. 12; no. 12; p. 1878
• Main Author: Yang, Yang
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.06.2024
• Subjects: Algorithms; all-capacities unbounded knapsack problem; branch and bound; Complexity; Diophantine equation; Dynamic programming; Fast Fourier transformations; Fourier transforms; Frobenius problem; Knapsack problem; linear Diophantine equation; Linear programming; unbounded knapsack problem with bounded coefficients; Variables; United Kingdom
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math12121878

Benchmark instances for the unbounded knapsack problem are typically generated according to specific criteria within a given constant range R, and these instances can be referred to as the unbounded knapsack problem with bounded coefficients (UKPB). In order to increase the difficulty of solving these instances, the knapsack capacity C is usually set to a very large value. While current efficient algorithms primarily center on the Fast Fourier Transform (FFT) and (min,+)-convolution method, there is a simpler method worth considering. In this paper, based on the basic Unbounded-DP algorithm, we utilize a recent branch and bound (B&B) result and basic theory of linear Diophantine equation, and propose an improved Unbounded-DP algorithm with time complexity of O(R4) and space complexity of O(R3). Additionally, the algorithm can also solve the All-capacities unbounded knapsack problem within the complexity O(R4+C). In particular, the proof techniques required by the algorithm are primarily covered in the first-year mathematics curriculum, which is convenient for subsequent researchers to grasp.