Efficient exact sequential lifting algorithm for binary knapsack set

TL;DR

This paper introduces an exact sequential lifting algorithm for binary knapsack sets that improves computational efficiency and stability, especially for large-scale instances, by using a dominance list structure and reduction methods.

Contribution

It presents a novel lifting algorithm employing dominance lists and reduction techniques, enabling exact lifting for binary knapsack sets with non-integer weights and large capacities.

Findings

Outperforms dynamic programming in efficiency and stability

Handles non-integer weights effectively

Suitable for large-scale MIP problems

Abstract

Lifting is a crucial technique in mixed integer programming (MIP) for generating strong valid inequalities, which serve as cutting planes to improve the branch-and-cut algorithm. We first propose an exact sequential lifting algorithm for the binary knapsack set, which employs the dominance list structure to remove redundant storage and computation in the dynamic programming (DP) array. This structure preserves scale invariance and effectively handles constraints with non-integer coefficients. Then, a reduction method is developed for the lifting procedure under some conditions, further enhancing computational efficiency. Finally, numerical experiments demonstrate that the proposed algorithm outperforms DP with arrays in terms of both efficiency and stability, particularly for large-scale and large-capacity instances. Moreover, it enables exact sequential lifting for binary knapsack sets with non-integer weights and large capacities, making it directly applicable in modern MIP solvers.