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Efficient exact sequential lifting algorithm for binary knapsack set

Xintong Wang, Liang Chen, Yu-Hong Dai

TL;DR

An exact sequential lifting algorithm is proposed for the binary knapsack set, which employs the dominance list structure to remove redundant storage and computation in the dynamic programming (DP) array, which preserves scale invariance and effectively handles constraints with non-integer coefficients.

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.

Efficient exact sequential lifting algorithm for binary knapsack set

TL;DR

An exact sequential lifting algorithm is proposed for the binary knapsack set, which employs the dominance list structure to remove redundant storage and computation in the dynamic programming (DP) array, which preserves scale invariance and effectively handles constraints with non-integer coefficients.

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.
Paper Structure (22 sections, 4 theorems, 21 equations, 3 figures, 21 tables, 3 algorithms)

This paper contains 22 sections, 4 theorems, 21 equations, 3 figures, 21 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Motivation and main contributions
  4. Sequential lifting on binary knapsack set
  5. Sequential lifting with dominance list
  6. State and dominance list
  7. Dominance list iteration
  8. Reduction method
  9. Exact sequential lifting algorithm
  10. Numerial Results
  11. Comparison of the DL, DP and reduction method
  12. Testset
  13. Performance and impact of the number of item and capacity
  14. Performance and impact of $\lambda$ and capacity
  15. Performance of scale invariance
  16. ...and 7 more sections

Key Result

Lemma 1

For any two distinct states $(w',p')$ and $(w",p")$ in a dominance list $L_k$, either $w'<w"$ and $p'<p"$ or $w'>w"$ and $p'>p"$.

Figures (3)

  • Figure 1: Performance of $b$ for four settings under $n=1000,~\lambda = 0.6$
  • Figure 2: Performance of $n$ for four settings under $b=10^5,~\lambda = 0.6$
  • Figure 3: Performance of $\lambda$ for four settings under $n=3000,~b=10^4$

Theorems & Definitions (14)

  • Example 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Example 2
  • Example 3
  • ...and 4 more