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Lifting for the integer knapsack cover polyhedron

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Abstract

We consider the integer knapsack cover polyhedron which is the convex hull of the set consisting of n-dimensional nonnegative integer vectors that satisfy one linear constraint. We study the sequentially lifted (SL) inequality, derived by the sequential lifting from a seed inequality containing a single variable, and provide bounds on the lifting coefficients, which is useful in solving the separation problem of the SL inequalities. The proposed SL inequality is shown to dominate the well-known mixed integer rounding (MIR) inequality under certain conditions. We show that the problem of computing the coefficients for an SL inequality is \(\mathcal{N}\mathcal{P}\)-hard but can be solved by a pseudo-polynomial time algorithm. As a by-product of analysis, we provide new conditions to guarantee the MIR inequality to be facet-defining for the considered polyhedron and prove that in general, the problem of deciding whether an MIR inequality defines a facet is \(\mathcal{N}\mathcal{P}\)-complete. Finally, we perform numerical experiments to evaluate the performance and impact of using the proposed SL inequalities as cutting planes in solving mixed integer linear programming problems. Numerical results demonstrate that the proposed SL cuts are much more effective than the MIR cuts in terms of strengthening the problem formulation and improving the solution efficiency. Moreover, when applied to solve random and real application problems, the proposed SL cuts demonstrate the benefit in reducing the solution time.

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Notes

  1. Here we remark that a new stopping criterion is introduced in Chen and Dai [14] for the linear time algorithm by Pochet and Wolsey [46] and can reduce the number of iterations. The improved algorithm is then applied in Chen and Dai [14] to derive a combinatorial polynomial time algorithm for the separation problem of continuous knapsack polyhedron with divisible capacities considered in [56].

  2. By \(\frac{r}{\alpha _{\min }} \in {\mathbb {Z}}\), we have \(\frac{rk_i+r}{\alpha _{\min }} \in {\mathbb {Z}}\) for all \(i \in L\).

  3. Here rows refers to constraints or valid inequalities of an MILP problem.

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Acknowledgements

The authors thank the associate editor and two anonymous referees for their insightful comments that helped significantly improve the presentation of this paper. The work of W.-K. Chen was supported in part by the Chinese NSF (Grant No. 12101048) and Beijing Institute of Technology Research Fund Program for Young Scholars. The work of Y.-H. Dai was supported in part by the Chinese NSF (Grant Nos. 12021001, 11991021, 11991020, and 11971372), the National Key R &D Program of China (Grant Nos. 2021YFA1000300 and 2021YFA1000301), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA27000000).

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Authors and Affiliations

  1. School of Mathematics and Statistics/Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, China

    Wei-Kun Chen

  2. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Liang Chen & Yu-Hong Dai

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

    Liang Chen & Yu-Hong Dai

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  1. Wei-Kun Chen
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  2. Liang Chen
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Correspondence to Yu-Hong Dai.

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Appendix: detailed computational results

Appendix: detailed computational results

See the Tables 6, 7 and 8.

Table 6 Comparison of SL, MIR, and SL+MIR
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Table 7 Comparison of SCIP and SCIP+SL
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Table 8 Comparisons of different selections of parameter \(\texttt {MAXAGGR}\)
Full size table

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Chen, WK., Chen, L. & Dai, YH. Lifting for the integer knapsack cover polyhedron. J Glob Optim 86, 205–249 (2023). https://doi.org/10.1007/s10898-022-01252-x

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