Mathematical Programming

, Volume 113, Issue 1, pp 15-37

First online:

Polyhedral properties for the intersection of two knapsacks

  • Quentin LouveauxAffiliated withDepartment of Mathematics/IMO, Otto-von-Guericke-Universität Magdeburg Email author 
  • , Robert WeismantelAffiliated withDepartment of Mathematics/IMO, Otto-von-Guericke-Universität Magdeburg

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We address the question to what extent polyhedral knowledge about individual knapsack constraints suffices or lacks to describe the convex hull of the binary solutions to their intersection. It turns out that the sign patterns of the weight vectors are responsible for the types of combinatorial valid inequalities appearing in the description of the convex hull of the intersection. In particular, we introduce the notion of an incomplete set inequality which is based on a combinatorial principle for the intersection of two knapsacks. We outline schemes to compute nontrivial bounds for the strength of such inequalities w.r.t. the intersection of the convex hulls of the initial knapsacks. An extension of the inequalities to the mixed case is also given. This opens up the possibility to use the inequalities in an arbitrary simplex tableau.