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A Note on Split Rank of Intersection Cuts

Santanu Dey (santanu.dey***at***uclouvain.be)

Abstract: In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the lattice-free convex set that is used to generate the intersection cut is constructed. We call this subset the restricted lattice-free set. It is then shown that $\lceil \textrm{log}_2 (l)\rceil$ is a lower bound on the split rank of the intersection cut, where $l$ is the number of integer points lying on the boundary of the restricted lattice-free set satisfying the condition that no two points lie on the same facet of the restricted lattice-free set. The use of this result is illustrated to obtain a lower bound of $\lceil \textrm{log}_2( n +1) \rceil$ on the split rank of $n$-row mixing inequalities.

Keywords: Split Rank , Intersection Cuts, Mixing Inequalities, Constant Capacity Lot-Sizing Problem

Category 1: Integer Programming ((Mixed) Integer Linear Programming )

Category 2: Integer Programming (Cutting Plane Approaches )

Citation: CORE DP 56, 2008

Download: [PDF]

Entry Submitted: 09/22/2008
Entry Accepted: 09/22/2008
Entry Last Modified: 10/09/2008

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