Mathematical Programming

, Volume 113, Issue 1, pp 15–37

Polyhedral properties for the intersection of two knapsacks

Authors

    • Department of Mathematics/IMOOtto-von-Guericke-Universität Magdeburg
  • Robert Weismantel
    • Department of Mathematics/IMOOtto-von-Guericke-Universität Magdeburg
FULL LENGTH PAPER

DOI: 10.1007/s10107-006-0045-9

Cite this article as:
Louveaux, Q. & Weismantel, R. Math. Program. (2008) 113: 15. doi:10.1007/s10107-006-0045-9

Abstract

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.

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© Springer-Verlag 2006