# Multi-cover Inequalities for Totally-Ordered Multiple Knapsack Sets

@inproceedings{Pia2021MulticoverIF, title={Multi-cover Inequalities for Totally-Ordered Multiple Knapsack Sets}, author={Alberto Del Pia and Jeff T. Linderoth and Haoran Zhu}, booktitle={IPCO}, year={2021} }

We propose a method to generate cutting-planes from multiple covers of knapsack constraints. The covers may come from different knapsack inequalities if the weights in the inequalities form a totally-ordered set. Thus, we introduce and study the structure of a totally-ordered multiple knapsack set. The valid multi-cover inequalities we derive for its convex hull have a number of interesting properties. First, they generalize the well-known (1, k)configuration inequalities. Second, they are not…

## References

SHOWING 1-10 OF 26 REFERENCES

Aggregation-based cutting-planes for packing and covering integer programs

- Mathematics, Computer ScienceMath. Program.
- 2018

This paper examines the strength of cuts based on k different aggregation inequalities simultaneously, the so-called multi-row cuts, and shows that every packing or covering IP with a large integrality gap also has a large k-aggregation closure rank.

On the exact separation of mixed integer knapsack cuts

- Mathematics, Computer ScienceMath. Program.
- 2011

An algorithm to separate over the convex hull of a mixed integer knapsack set which exploits dominance relationships is developed, and its computations, which are performed in exact arithmetic, are surprising: In the vast majority of the instances in whichknapsack cuts yield bound improvements, MIR cuts alone achieve over 87% of the observed gain.

Solving Multiple Knapsack Problems by Cutting Planes

- Mathematics, Computer ScienceSIAM J. Optim.
- 1996

The inequalities that are described here serve as the theoretical basis for a cutting plane algorithm that is applied to practical problem instances arising in the design of main frame computers, in the layout of electronic circuits, and in sugar cane alcohol production.

A characterization of knapsacks with the max-flow- - min-cut property

- Computer Science, MathematicsOper. Res. Lett.
- 1992

Using a result of Seymour, a class of knapsack problems for which the clutter of minimal covers has the max-flow-min-cut property with respect to all right-hand sides implies that adding the minimal cover cuts to the problem is sufficient to guarantee an integer optimum for the linear programming relaxation.

Facet of regular 0–1 polytopes

- Mathematics, Computer ScienceMath. Program.
- 1975

A special class of prime implicants is described for regular functions and it is shown that for anyP in this class,F(P) consists of one facet of H, and this facet has 0–1 coefficients, and every nontrivial facet ofH with 0-1 coefficients is obtained from this class.

On the 0/1 knapsack polytope

- Mathematics, Computer ScienceMath. Program.
- 1997

The class ofweight inequalities is introduced, needed to describe the knapsack polyhedron when the weights of the items lie in certain intervals, and the properties of lifted minimal cover inequalities are extended to this general class of inequalities.

Lifted Cover Inequalities for 0-1 Integer Programs: Complexity

- Mathematics, Computer ScienceINFORMS J. Comput.
- 1999

There exists a class of 0-1 knapsack instances for which any branch-and-cut algorithm based on LCIs has to evaluate an exponential number of nodes to prove optimality.

Facets of the knapsack polytope

- Mathematics, Computer ScienceMath. Program.
- 1975

A necessary and sufficient condition is given for an inequality with coefficients 0 or 1 to define a facet of the knapsack polytope, i.e., of the convex hull of 0–1 points satisfying a given linear…

Canonical Cuts on the Unit Hypercube

- Mathematics
- 1972

In this paper we study some properties of the n-dimensional unit hypercube K. We define a distance function on the set V of vertices of K, and use it to construct a class of hyperplanes parallel to…

Knapsack polytopes: a survey

- Mathematics, Computer ScienceAnn. Oper. Res.
- 2020

This paper provides a comprehensive overview of knapsack polytopes, discussing basic polyhedral properties, (lifted) cover and other valid inequalities, cases for which complete linear descriptions are known, geometric properties for small dimensions, and connections to independence systems.