Mathematical Programming

, Volume 114, Issue 2, pp 207–234 | Cite as

Second-order cover inequalities



We introduce a new class of second-order cover inequalities whose members are generally stronger than the classical knapsack cover inequalities that are commonly used to enhance the performance of branch-and-cut methods for 0–1 integer programming problems. These inequalities result by focusing attention on a single knapsack constraint in addition to an inequality that bounds the sum of all variables, or in general, that bounds a linear form containing only the coefficients 0, 1, and –1. We provide an algorithm that generates all non-dominated second-order cover inequalities, making use of theorems on dominance relationships to bypass the examination of many dominated alternatives. Furthermore, we derive conditions under which these non-dominated second-order cover inequalities would be facets of the convex hull of feasible solutions to the parent constraints, and demonstrate how they can be lifted otherwise. Numerical examples of applying the algorithm disclose its ability to generate valid inequalities that are sometimes significantly stronger than those derived from traditional knapsack covers. Our results can also be extended to incorporate multiple choice inequalities that limit sums over disjoint subsets of variables to be at most one.


Integer programming Knapsack cover inequalities 0–1 Pre-processing Nested cuts Surrogate constraints Facets 

Mathematics Subject Classification

90C10 90C27 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.University of ColoradoBoulderUSA
  2. 2.Grado Department of Industrial and Systems EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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